1 Introduction

Discovery of Hawking radiation [1, 2] bridges quantum mechanics and general relativity theory, where it was predicted that the black hole emits thermal radiation like a black body with an entropy proportional to the black hole area [3, 4] and the temperature called the Hawking temperature that proportional to the surface gravity. Thermodynamics of Schwarzschild-AdS black hole established by Hawking-Page [5] demands a huge importance by researcher. The negative cosmological constant \(\Lambda \) is considered as the thermodynamic pressure \(P=-\frac{\Lambda }{8 \pi }\) and the black hole mass is regarded as chemical enthalpy [6,7,8,9]. Ads-black hole geometry and its thermodynamics has been studied by many authors [10,11,12,13,14,15,16].

The study of static black hole thermodynamics provides a significant arena in modern theoretical physics. Later on, dynamical black hole thermodynamics was studied on the utter trapping horizon as apparent horizon by Hayward et al. [17,18,19] and expressed the Einstein field equations by unified first law in the spherical symmetric space-time. The concept of black hole thermodynamics is extended to the thermodynamic study of FRW universe different gravity theories, namely they are Einstein gravity, Gauss–Bonnet gravity and Lovelock gravity by Cai and Kim [20, 21]. Further, Akbar and Cai [22,23,24] extended that study of the thermodynamics of FRW universe in scalar-tensor gravity, Gauss–Bonnet gravity, Lovelock gravity and f(R) gravity. Several black hole models like AdS black hole, Gauss–Bonnet black hole, Chaplygin black hole, Born–Infeld AdS black hole, Van der Waals Black Hole [25,26,27,28,29,30,31,32] had studied as heat engine and the work-done efficiency calculated. The efficiency of work done due to the transformation of heat into mechanical energy in a heat engine depends on the different choices of thermodynamic process. The idea that a universe is assumed as a thermodynamic system and use that system as a heat engine is done in a recent work by Debnath [33] where the author assumed the FRW universe as a thermodynamic system and calculated the thermodynamic quantities on the dynamic apparent horizon and relate in terms of thermodynamic system entropy on the horizon and used the universe as a heat engine. He also studied the more general thermodynamic properties and heat engine for FRW universe [34]. A few more studies are done by some other authors, namely by Pilot [35], a study of polytropic Carnot heat engine by the author Askin [36]. The Amended Friedmann–Robertson–Walker (AFRW) metric was introduced by Hunter [37] to study the redshift. Amir et al. [38] have assumed the AFRW universe model with Ricci Dark Energy in Chern–Simons modified gravity. Further, Moradi et al. [39] have also considered Rényi Dark Energy model in the AFRW universe. All the studies inspire us to consider a non-flat universe, namely the AFRW universe, where the scale factor plays an important role in the expanding evaluation of the universe as a thermodynamic heat engine.

The following is how the paper is organized: In the second section, we examine the AFRW universe’s first law of thermodynamics using the cosmological constant as the thermodynamic pressure. In Sect. 3, we obtain the entropy using unified first law. The temperature, entropy, and volume are evaluated on the apparent horizon. We derive the thermodynamic quantities enthalpy, Gibb’s Helmholtz’s free energies, the specific heat capacity of the universe, Joule–Thomson coefficient, and inversion pressure and temperature expressed in terms of apparent horizon surface area. In Sect. 4, the AFRW universe is assumed to be a heat engine for which we study the Carnot engine and a new engine with work done and its efficiency. In Sect. 5, we finally write the overall work’s conclusions.

2 Unified first law and thermodynamic equation of state with cosmological constant

The space-time metric of homogeneous, isotropic and non-flat Amended Friedmann Robertson Walker (AFRW) Universe is [37]

$$\begin{aligned} ds^2=-a^2(t)dt^2+a^2(t)\left( \frac{dr^2}{1-kr^2}+r^2 d \Omega _{2}^{2}\right) \end{aligned}$$
(1)

where a(t), the scale factor plays an important role to describe the expanding nature of the universe. Flat, open and closed model of the Universe correspond to the values of \(k=0, -1, 1\). With an energy density of \(\rho \) and a pressure of p, the energy-momentum tensor \(T_{ij}\) for the perfect fluid is satisfied

$$\begin{aligned} T_{ij}=(\rho +p)u_iu_j+pg_{ij} \end{aligned}$$
(2)

The four velocity of the \(u_i\) satisfying the equation \(u_iu^i=-1\) and \(u^i \nabla _j u_i=0\). Expressing the metric (1) in the form [40]

$$\begin{aligned} ds^2=h_{ij}dx^idx^i+{\tilde{r}}^2(d\theta ^2+sin^2\theta d\phi ^2) \end{aligned}$$
(3)

we get \(x^0=t, x^1=r, {\tilde{r}}=a(t) r \) and \(h_{ij}=diag(-1,\frac{a^2}{1-kr^2})\). AFRW universe is regarded as thermodynamical system where the apparent horizon radius \({\tilde{r}}\) satisfies the relation \(h^{ij}\frac{\partial {\tilde{r}}}{\partial x^i}\frac{\partial {\tilde{r}}}{\partial x^j}=0\). The apparent horizon radius \(\tilde{r_h}\) for AFRW universe is obtained as

$$\begin{aligned} {\tilde{r}}_h=\frac{a}{\sqrt{H^2+k}} \end{aligned}$$
(4)

Differentiating (4) with respect to time t gives

$$\begin{aligned} \dot{{\tilde{r}}}_h={\tilde{r}}_h H \left( 1-\frac{{\dot{H}}{\tilde{r}}^2_h}{a^2}\right) \end{aligned}$$
(5)

where \(H=\frac{{\dot{a}}}{a}\) be the Hubble parameter. Numerous academic works demonstrate the relationship between the dynamical apparent horizon, gravitational entropy, and surface gravity, with the surface gravity on the apparent horizon being determined by

$$\begin{aligned} \kappa = \frac{1}{2\sqrt{-h}}\frac{\partial }{\partial x^i}\left( \sqrt{-h} h^{ij}\frac{\partial {\tilde{r}}}{\partial x^j} \right) \end{aligned}$$
(6)

where \(h=det(h_{ij})\). The apparent horizon temperature T and surface gravity for the AFRW universe are obtained as

$$\begin{aligned} \kappa = -\frac{a^2}{{\tilde{r}}^2_h}\left( 1-\frac{\dot{{\tilde{r}}}_h}{2 {\tilde{r}}_h H }\right) \end{aligned}$$
(7)

and

$$\begin{aligned} T=\frac{|\kappa |}{2 \pi }= \frac{a^2}{2 \pi {\tilde{r}}_h}\left( 1-\frac{\dot{{\tilde{r}}}_h}{2 {\tilde{r}}_h H }\right) \end{aligned}$$
(8)

Now since \(\dot{{\tilde{r}}}_h={\tilde{r}}_h H\), then the Hawking’s temperature become

$$\begin{aligned} T=\frac{a^2}{4\pi {\tilde{r}}_h } \end{aligned}$$
(9)

The unified first law of thermodynamics of a dynamic black hole proposed by Hayward [17, 41] gives the change of energy (dE) inside the dynamic apparent horizon expressed by

$$\begin{aligned} dE=A \Psi + W dV \end{aligned}$$
(10)

where W and \(\Psi \) are respectively the work density function due to the dynamic nature of apparent horizon and the total energy flow through the surface. Furthermore the surface area A and the thermodynamic volume V within a sphere with apparent horizon radius \( {\tilde{r}}_h\) is satisfying the following equations

$$\begin{aligned} A=4\pi {\tilde{r}}^2_h \end{aligned}$$
(11)

and

$$\begin{aligned} V=\frac{4}{3}\pi {\tilde{r}}^3_h \end{aligned}$$
(12)

Now the work density function W [17, 40,41,42] is given by

$$\begin{aligned} W=-\frac{1}{2} T^{ij} h_{ij}=\frac{1}{2} \left( \rho -p \right) \end{aligned}$$
(13)

and the energy flow vectors through the apparent horizon is

$$\begin{aligned} \Psi _i=h^{jk}T_{ik}\frac{\partial {\tilde{r}}}{\partial x^j}+W\frac{\partial {\tilde{r}}}{\partial x^i} \end{aligned}$$
(14)

For our AFRW universe, the components of energy-supply vector are

$$\begin{aligned} \Psi _t=-\frac{1}{2}H {\tilde{r}} \left( \rho +p \right) , \Psi _r=\frac{a}{2} \left( \rho +p \right) \end{aligned}$$
(15)

Hence the energy flux become

$$\begin{aligned} \Psi =\Psi _i dx^i=-\left( \rho +p \right) H {\tilde{r}} dt+\frac{1}{2}\left( \rho +p \right) d{\tilde{r}} \end{aligned}$$
(16)

So the change of energy dE (10) is obtained as

$$\begin{aligned} dE=A \Psi + W dV= -A\left( \rho +p \right) H {\tilde{r}}_h dt+A \rho d{\tilde{r}}_h \end{aligned}$$
(17)

Now if \(\delta Q\) represents the heat flow through the dynamic apparent horizon, then equating that to change of energy dE through the apparent horizon in the time interval dt [41], we found

$$\begin{aligned} \delta Q=-dE=A \left( \rho +p \right) H {\tilde{r}}_h dt \end{aligned}$$
(18)

Further on the apparent horizon due to the Clausius first law of thermodynamics we get

$$\begin{aligned} \delta Q=T dS=A \left( \rho +p \right) H {\tilde{r}}_h dt \end{aligned}$$
(19)

For the AFRW universe filled with fluid with energy density \(\rho \) and pressure p, the modified Friedmann equations in presence of cosmological constant in Einstein’s gravity is

$$\begin{aligned} \frac{H^2+k}{a^2}=\frac{8\pi G}{3} \rho +\frac{\Lambda }{3} \end{aligned}$$
(20)

and

$$\begin{aligned} {\dot{H}}=-\frac{4 \pi G a^2}{3} \left( \rho +3 p\right) \end{aligned}$$
(21)

Energy conservation equation is

$$\begin{aligned} {\dot{\rho }}+3H\left( \rho +p\right) =0 \end{aligned}$$
(22)

using the above equation the following are obtained

$$\begin{aligned} \rho =\frac{3}{8 \pi G {\tilde{r}}^2_h} -\frac{\Lambda }{8 \pi G} \end{aligned}$$
(23)

and

$$\begin{aligned} p=\frac{\Lambda }{24 \pi G } -\frac{1}{8 \pi G {\tilde{r}}^2_h} \end{aligned}$$
(24)

where \({\tilde{r}}_h^2=\frac{A}{4 \pi }\).

3 Thermodynamic quantities

Incorporating the cosmological constant \(\Lambda \) as the thermodynamic pressure \(P=-\frac{\Lambda }{8 \pi G}\) in the thermodynamics of AFRW universe (23) and (24) yield

$$\begin{aligned} \rho =\frac{3}{2 A G }+P \end{aligned}$$
(25)

and

$$\begin{aligned} p=-\frac{P}{3}-\frac{1}{2 A G } \end{aligned}$$
(26)

Since, \(\Lambda \) is negative, implies \(\rho >0\). Further the volume inside the apparent horizon and the temperature on the dynamic apparent horizon are expressed in terms apparent horizon surface area A as

$$\begin{aligned} V=\frac{1}{6 \sqrt{\pi }} A^{3/2} \end{aligned}$$
(27)

and

$$\begin{aligned} T=\frac{b G \sqrt{A}}{\sqrt{ \pi }(3+2 A G P)} \end{aligned}$$
(28)

Now using (25), (26) in (19) and integrating we can obtain the entropy S as

$$\begin{aligned} S=\frac{A P}{b G}+\frac{A^2 P^2}{6 b}+\frac{3 \log A}{4 b G^2}+S_0 \end{aligned}$$
(29)

where, \(S_0\) being the integrating constant and the constant b satisfies \(a^2=\frac{2AG b}{(3+2AGP)}\). The Eq. (29) establish the relation between the entropy and the surface area of the dynamic apparent horizon and hence with the apparent horizon radius. In Fig. 1a–c we plot the T, V and S respectively with the variation of surface area A. It is found that temperature, Volume inside the apparent horizon and entropy are increasing as the value of the surface area is increasing.

Further, we derive the thermodynamic quantities of our assumed thermodynamic system, the AFRW universe. Enthalpy is one of the thermodynamic quantity and an intensive property of a system in thermodynamics and defined by \(H=U+PV\), U represents the internal energy of the system. Using first law of thermodynamics for AFRW universe, the enthalpy function can be derived as

$$\begin{aligned} d\mathcal {H}=TdS+VdP \end{aligned}$$
(30)

gives

$$\begin{aligned} \mathcal {H}=-\int \frac{1}{2\sqrt{\pi }}A^{3/2}dp+\frac{\sqrt{A}(9+AGP)}{9G\sqrt{\pi }}+\mathcal {H}_0 \end{aligned}$$
(31)

where \(\mathcal {H}_0\) be integrating constant. Then the expression for Gibb’s free energy [43] become

$$\begin{aligned} \mathcal {G}= & {} \mathcal {H}-TS \nonumber \\= & {} -\int \frac{1}{2\sqrt{\pi }}A^{3/2}dp \nonumber \\{} & {} +\frac{\sqrt{A}(108{+}48AGP{+}2 A^2 G^2 P^2-27 \log A {-}36b G^2 S_0)}{36G\sqrt{\pi }(3{+}2AGP)}\nonumber \\{} & {} +\mathcal {H}_0 \end{aligned}$$
(32)

and Helmholtz’s free energy [43] is expressed as

Fig. 1
figure 1

a Plot of temperature with respect to the apparent horizon surface area. b Plot of volume inside the apparent horizon with respect to the apparent horizon surface area. c Plot of entropy with respect to the apparent horizon surface area

Full size image
$$\begin{aligned} \mathcal {F}= & {} \mathcal {G}-PV=-\int \frac{1}{2\sqrt{\pi }}A^{3/2}dp\nonumber \\{} & {} +\frac{\sqrt{A}(108+48AGP+2 A^2 G^2 P^2-27 \log A -36b G^2 S_0)}{36G\sqrt{\pi }(3+2AGP)} \nonumber \\{} & {} -\frac{A^{3/2}P}{6 \sqrt{\pi }}+\mathcal {H}_0 \end{aligned}$$
(33)

Here we plot \(\mathcal {H}\), \(\mathcal {G}\) and \(\mathcal {F}\) in Fig. 2a–c with the change of area of the apparent horizon A. The figures depicts that Enthalpy increases and on the other side Gibb’s free energy and Helmholtz’s free energy both decreases with increase of apparent horizon surface area. Next for AFRW universe the specific heat capacity \(C_P\) [8] is obtained as

$$\begin{aligned} C_P=T\left( \frac{\partial S}{\partial T}\right) _P= \frac{(3+2AGP)^3}{6 b G^3 (3-2AGP)} \end{aligned}$$
(34)

and the coefficient of thermal expansion is given by

$$\begin{aligned} \alpha =\frac{1}{V}\left( \frac{\partial V}{\partial T}\right) _P=\frac{3\sqrt{\pi }(3+2AGP)^2}{b G \sqrt{A}(3-2AGP)} \end{aligned}$$
(35)
Fig. 2
figure 2

a Plot of enthalpy with respect to the apparent horizon surface area. b Plot of Gibb’s free energy inside the apparent horizon with respect to the apparent horizon surface area. c Plot of Helmholtz’s free energy with respect to the apparent horizon surface area

Full size image
Fig. 3
figure 3

Heat engine P–V diagram

Full size image

The isothermal compressibility [44] for the universe is

$$\begin{aligned} K_T=-\frac{1}{V} \left( \frac{\partial V}{\partial P}\right) _T=\frac{6AG}{(2AGP-3)} \end{aligned}$$
(36)

Now, since Eq. (26) is showing \(p<0\), hence (36) implies \(p<-\frac{1}{A G}\) for the positive value of \(K_T\). The Joule–Thomson coefficient (\(\mu \)) is defined as the change of temperature with respect to the change in pressure while the system enthalpy remains constant [46] means that \(\mu \) can be determined as the slope of the isenthalpic curve given by [45,46,47]

$$\begin{aligned} \mu =\left( \frac{\partial T}{\partial P}\right) _\mathcal {H} \end{aligned}$$
(37)

\(\mu \) determines the cooling and heating nature of the universe corresponding to \(\mu >0\) or \(\mu <0\) respectively. The Joule–Thomson coefficient also can be expressed as

$$\begin{aligned} \mu =\frac{1}{C_P}\left[ T\left( \frac{\partial V}{\partial T}\right) _P-V\right] \end{aligned}$$
(38)

or,

$$\begin{aligned} \mu =\frac{1}{S}\left[ P\left( \frac{\partial V}{\partial P}\right) _\mathcal {H}+2V\right] \end{aligned}$$
(39)

which gives for our AFRW universe

$$\begin{aligned} \mu =\frac{2bG^2 A^{3/2}(3+4AGP)}{\sqrt{\pi }(3+2AGP)^3} \end{aligned}$$
(40)
Fig. 4
figure 4

a Plot of specific heat with respect to the apparent horizon surface area. b Plot of coefficient of thermal expansion with respect to the apparent horizon surface area. c Plot of Joule–Thomson coefficient with respect to the apparent horizon surface area

Full size image
Fig. 5
figure 5

a Plot of isothermal compressibility with respect to the apparent horizon surface area. b Plot of inversion temperature with respect to the apparent horizon surface area. c Plot of inversion pressure with respect to the apparent horizon surface area

Full size image
Fig. 6
figure 6

a Plot of work done efficiency of Carnot heat engine in AFRW universe with respect to the apparent horizon surface area. b Plot of work done efficiency of new heat engine in AFRW universe with respect to the apparent horizon surface area

Full size image

Since here we found \(\mu >0\), so the Amended FRW universe is showing a cooling behaviour of the universe.

Now putting \(\mu =0\) in (38) gives the inversion temperature obtained as

$$\begin{aligned} T_{inv}=V \left( \frac{\partial T}{\partial V}\right) _P=\frac{\sqrt{A} b G (3-2AGP)}{3 \sqrt{\pi } (3+2AGP)^2} \end{aligned}$$
(41)

and putting \(\mu =0\) in (39) gives the inversion pressure as

$$\begin{aligned} P_{inv}=-\frac{2V}{\left( \frac{\partial V}{\partial P}\right) _\mathcal {H}}=\frac{32}{3 A G} \end{aligned}$$
(42)

Further, three thermodynamic quantities \(C_P\), \(\alpha \) and \(\mu \) are graphically represented in Fig. 4a–c and other three quantities namely \(K_T\), \(T_{inv}\) and \(P_{inv}\) are in Fig. 5a–c with respect to A. Here we see that \(C_P\) and \(\alpha \) both are decreasing while Joule–Thomson coefficient \(\mu \) is increasing with respect A and \(\mu \) being positive showing cooling nature of AFRW universe. For our AFRW universe \(K_T\) and inversion pressure both are decreasing but the inversion temperature is increasing with A.

4 AFRW universe as heat engine

A heat engine is a thermodynamical system where heat flows from a heat source to a heat sink and converts the thermal energy to usable mechanical energy. A reversible cycle termed as Carnot cycle to describe the thermodynamics of the heat engine in which we assume that \(T_H\) and \(T_C\) are respectively the temperatures of the heat source and a heat sink. Carnot cycle consist of two isothermal processes and two adiabetc process as shown in P–V diagram (Fig. 3) [48]. Along the upper isotherm the heat flow inside the system is \( Q_H=T_H \Delta S_{1 \longrightarrow 2}=T_H(S_2-S_1)\) and along the lower isotherm the heat leave the system \(Q_C=T_C\Delta S_{3 \longrightarrow 4}=T_C(S_3-S_4)\). Further in the AFRW universe the relation between entropy and the apparent surface area is connected by the relation

$$\begin{aligned} S_i=\frac{A_i P}{b G}+\frac{A_i^2 P^2}{6 b}+\frac{3 \log A_i}{4 b G^2}+S_0,~~~~i=1, 2, 3, 4\nonumber \\ \end{aligned}$$
(43)

where \(A_i=\left( 6\sqrt{\pi }\right) ^{3/2}V_i^{2/3},i=1, 2, 3, 4\). Thus the total work done in AFRW universe as heat engine is \(W=Q_H-Q_C\) and the efficiency of the Carnot heat engine \(\eta _{car}=\frac{W}{Q_H}=1-\frac{Q_C}{Q_H}\). Now for Carnot cycle as \(V_1=V_4\) and \(V_2=V_3\), maximum efficiency become \(\left( \eta _{car}\right) _{Max}=1-\frac{T_C}{T_H}\) and hence \(0<\left( \eta _{car}\right) _{Max}<1\) as \(T_C<T_H\).

Now for the new heat engine as described by Johnson [48], where the cycle consists with two isobar ans two isochores as shown in Fig. 3 and heat flows along the two isobars. For this heat engine the total work done become

$$\begin{aligned} W=\Delta P_{4 \longrightarrow 1} \Delta V_{1 \longrightarrow 2}=(P_1-P_4)(V_2-V_1) \end{aligned}$$
(44)

where, \(P_i\) and \(V_i\) indicates the fluid pressure and volume at i-stage in the cycle. Further along the upper isobar total heat iside the system is given by

$$\begin{aligned} Q_H= & {} \int _{T_1}^{T_2} C_P(P_1,T)dT=\frac{\sqrt{A_2} (2A_2GP_1+9)}{18 G \sqrt{\pi }}\nonumber \\{} & {} -\frac{\sqrt{A_1} (2A_1GP_1+9)}{18 G \sqrt{\pi }} \end{aligned}$$
(45)

and the total heat leave by the system along the lower isobar is given by

$$\begin{aligned} Q_c= & {} \int _{T_3}^{T_4} C_P(P_4,T)dT=\frac{\sqrt{A_4} (2A_4GP_4+9)}{18 G \sqrt{\pi }}\nonumber \\{} & {} -\frac{\sqrt{A_3} (2A_3GP_4+9)}{18 G \sqrt{\pi }} \end{aligned}$$
(46)

where, \(A_i=\frac{b^2\,G^2-12\,G P \pi T_i^2 +b G \sqrt{G(b^2\,G-24 P \pi T_i^2)}}{8\,G^2 P^2 \pi T_i^2}, i=1, 2, 3, 4\) and \(P=P_1\) or \(P=P_4\) respectively for (45) and (46). Now for real and positive \(A_i\) implies \(T_i^2<\frac{b^2 G}{12 \pi P_i}\), for \(i=1, 2, 3, 4\).

Finally the thermal efficiency for AFRW universe in the new engine becomes

$$\begin{aligned} \eta _{ne}=\frac{W}{Q_H}=\frac{3 G (A_1+A_2+\sqrt{A_1 A_2})(P_1-P_4)}{9+2 G P_1(A_1+A_2+\sqrt{A_1 A_2})} \end{aligned}$$
(47)

Here, we assume the AFRW universe as heat engine and two engine models Carnot and a new Heat engine, as shown in Fig. 3 is studied. The efficiencies \(\eta _{car}\) and \(\eta _{ne}\) of the work done in conversion of heat to mechanical energy are graphically shown in Fig. 6a and b.

5 Discussions and conclusions

A homogeneous, non-flat, isotropic AFRW universe with a negative cosmological constant, which is treated as that thermodynamic pressure, is considered as a thermodynamic system. Thermodynamic quantities like temperature, entropy, volume inside the apparent horizon, enthalpy, Gibb’s and Helmholtz’s free energies, specific heat capacity of the universe, Joule–Thomson coefficient, inversion temperature and inversion pressure are also calculated and expressed in terms of area of the dynamic apparent horizon surface. Also those quantities are plot in Figs. 1, 2, 4 and 5. Positive correlations of temperature, volume and entropy with the apparent horizon area are shown in Fig. 1a–c. Further, Fig. 2a–c depicts respectively enthalpy, Gibb’s free energy and Helmholtz’s free energy with the variation of the dynamic apparent horizon where we found that enthalpy of the universe increases, but both the Gibb’s and Helmhltz’s free energies decrease with A implies that the work done of the thermodynamic system is increasing in the evolution of AFRW universe. Additionally, Fig. 4c plots the Joule–Thomson coefficient. It is evident from this that \(\mu >0\) indicates the cooling nature of the AFRW universe, which is consistent with the FRW universe because the FRW universe also exhibits a positive Joule–Thomson coefficient, indicating a cooling nature [33, 34]. Also, the Specific heat with constant pressure (Fig. 4a) and coefficient of thermal expansion (Fig. 4b) are plotted, and they are showing a negative correlation with the surface area at the apparent horizon. A low propensity of thermal expansion with the temperature is seen here. Moreover, in Fig. 5a, isothermal compressibility is plotted for the negative thermodynamic pressure and shows a decreasing nature while the dynamic apparent horizon area increases. The inversion curve for temperature and pressure is the locus for which the Joule–Thomson coefficient \(\mu =0\). So, the inversion curve divides the \(T-P\) plane into two regions, one for which \(\mu >0\) shows the cooling universe and \(\mu <0\) shows the heating universe. Here, Fig. 5b and c give the inversion temperature and inversion pressure curve, respectively, with the variation of the area of the apparent surface. The inversion temperature increases, but the inversion pressure decreases in the increase of the surface area of the apparent horizon.

Heat flows from the heat source to the heat sink in a heat engine when a part is being transformed into mechanical energy. The heat engine therefore operates in a cycle. The heat engine’s efficiency can be defined as the ratio of the total work done with the heat flow within the system. This provides an indication of how well the engine converts thermal energy to mechanical energy. A determination of the actual state of the path varies the engine cycle of the FRW universe [33, 34]. Similar to the FRW universe, the AFRW universe is regarded as a heat engine because the heat presents in the universe, although the universe behaves like cooling nature. So heat flows from heat source to heat sink and the cycle is possible for AFRW universe. Here two heat engine models are studied for the AFRW universe, for which the cycles are presented in Fig. 3. In the Carnot heat engine, the reversible cycle is formed with two isothermal processes from stage \(1 \xrightarrow {} 2\) (temperature keeping \(T_H\)) and from stage \(3\xrightarrow {}4\) (temperature keeping \(T_Q\)) and two adiabatic processes from stage \(2\xrightarrow {}3\) and stage \(4\xrightarrow {}1\). In the new engine model [48] the isotherms are replaced by two isobars and adiabatic processes are by two isochoric processes (the cycles are shown in Fig. 3. Further, the work-done efficiencies are studied for the Carnot engine (\(\eta _{car}\)) and the new engine (\(\eta _{ne}\)) graphically. Figure 6a and b are the plots of work-done efficiencies \(\eta _{car}\) and \(\eta _{ne}\) respectively for Carnot engine and new engine with the variation of the apparent horizon surface area in the AFRW universe. In Fig. 6a, we assume \(P_1=10, P_2=8, P_3=4, P_4=3, b=0.01\) and \(\eta _{car}(=1-\frac{Q_C}{Q_H})\) with respect to the simultaneous variation of \(A_1\) and \(A_4\), respectively, are the area of the surface of the apparent horizon at stage 1 and at stage 4, \(P_i (i=1, 2, 3, 4)\) corresponds to the fluid pressure at \(i-\)stage, \(Q_H\) and \(Q_C\) are respectively the heat flow inside the system and leave the system. Also in Fig. 6b, where we plot \(\eta _{ne}\) with the variation of \(A_1\) and \(A_4\) and considering \(P_1=P_2=10, P_3=P_4=3\). Figure 6a depicts a different behaviour initially, and then it gets a constant variation with respect to \(A_1\) and \(A_4\). On the other hand, Fig. 6b indicates that \(\eta _{ne}\) starts with a higher value than \(\eta _{car}\) and strictly increases and the maximum value reached to 1. So, the work done efficiency by the conversion of thermal energy to mechanical energy, while AFRW universe is regarded as the new engine is higher than the case while AFRW universe is regarded as a Carnot engine.