The temperature of a Schwarzschild black hole can be derived heuristically using the standard HUP
$$\begin{aligned} \sigma _p(\Delta x)\sim \frac{\pi \hbar }{\Delta x}. \end{aligned}$$
(34)
Setting the uncertainty in position equal to the Schwarzschild radius \(r_s\) with the black hole mass M (in fact, \(\Delta x\) is basically the radius, so the real uncertainty is twice this value), one can relate the Hawking temperature to the standard deviation of momentum as
$$\begin{aligned} T_H^{(0)}=\frac{c}{k_B}\frac{\sigma _p(r_s)}{4\pi ^2}=\frac{c \hbar }{k_B}\frac{1}{4\pi r_s}. \end{aligned}$$
(35)
Using the first law of thermodynamics
$$\begin{aligned} \frac{\mathrm {d}S}{\mathrm {d}E}=T^{-1} \end{aligned}$$
(36)
with
$$\begin{aligned} E=2 r_s \frac{c^4}{4G} = r_s F_{max} , \end{aligned}$$
(37)
where \(F_{max}\) is the maximum force [56,57,58], the entropy can be integrated to give
$$\begin{aligned} S^{(0)}_{BH}=\frac{k_B c^3}{\hbar G}\pi r_s^2=\frac{k_B A_s}{4 l_p^2} \end{aligned}$$
(38)
where \(A_s\) and \(l_p\) are the area of the Schwarzschild horizon and the Planck length respectively.
Taking into account the uncertainty relations obtained in Sects. 3 and 4, the departure from the standard Bekenstein entropy and Hawking temperature can be calculated. This will be done for the asymptotic form of the EUP with horizons and the exact relations (18) and (29) obtained above.
Asymptotic form
As we can conclude from the previous results (19) and (31), the asymptotic form of the EUP for a background space-time which contains a horizon of radius \(r_{hor}\) reads
$$\begin{aligned} \sigma _p\sim \frac{\pi \hbar }{\Delta x}\left( 1+\beta _0 \frac{\Delta x^2}{r_{hor}^2} + O[(r_s/r_{hor})^4] \right) , \end{aligned}$$
(39)
and leads to the Hawking temperature
$$\begin{aligned} T_{H,as}=T_H^{(0)}\left( 1+\beta _0 \frac{r_s^2}{r_{hor}^2} + O[(r_s/r_{hor})^4] \right) \end{aligned}$$
(40)
which yields an entropy
$$\begin{aligned} S_{BH,as}&=\frac{ \pi k_B r_{hor}^2}{\beta _0 l_p^2}\log \left( 1+\beta _0\frac{r_s^2}{r_{hor}^2} + O[(r_s/r_{hor})^4] \right) \end{aligned}$$
(41)
$$\begin{aligned}&\simeq S_{BH}^{(0)}\left( 1-\frac{\beta _0}{2}\frac{r_s^2}{r_{hor}^2}+ O[(r_s/r_{hor})^4]\right) \end{aligned}$$
(42)
$$\begin{aligned}&\simeq S_{BH}^{(0)}\left( 1-\frac{\beta _0}{2}\frac{S_{BH}^{(0)}}{S_{hor}}+ O\left[ \left( \frac{S_{BH}^{(0)}}{S_{hor}}\right) ^2\right] \right) , \end{aligned}$$
(43)
where the horizon entropy of the background spacetime is equal to
$$\begin{aligned} S_{hor}=\frac{\pi k_B r_{hor}^2}{l_p^2}. \end{aligned}$$
(44)
Recall that \(\beta _0 < 0\) for given spacetimes, so the total entropy of the black hole is increased - this is an effect of taking canonical corrections due to some thermal fluctuations [47,48,49].
Rindler space
Applying the exact relation (18), the Hawking temperature of an accelerated black hole reads
$$\begin{aligned} T_{H,R}=\frac{\hbar \alpha }{8 \pi c k_B}\left( \sqrt{1+\frac{\alpha r_s}{2c^2}}-\sqrt{1-\frac{\alpha r_s}{2c^2}} \right) ^{-1} \end{aligned}$$
(45)
which leads to the entropy
$$\begin{aligned} S_{BH,R}{=}\frac{16\pi k_B}{3 l_p^2}\frac{c^4}{\alpha ^2}\left[ \left( 1+\frac{\alpha r_s}{2c^2}\right) ^{3/2}{+}\left( 1-\frac{\alpha r_s}{2c^2}\right) ^{3/2}\right] +S_0 \end{aligned}$$
(46)
with the integration constant \(S_0\)
$$\begin{aligned} S_0=-\frac{32\pi k_B}{3 l_p^2}\frac{c^4}{\alpha ^2} \end{aligned}$$
(47)
chosen for proper normalization (\(S_{BH,R}(r_s=0)=0\)). Thus, the entropy becomes
$$\begin{aligned} S_{BH,R}=\frac{16\pi k_B}{3 l_p^2}\frac{c^4}{\alpha ^2}\left[ \left( 1+\frac{\alpha r_s}{2c^2}\right) ^{3/2}+\left( 1-\frac{\alpha r_s}{2c^2}\right) ^{3/2}-2\right] . \end{aligned}$$
(48)
This entropy change encodes the entire non-perturbative influence of the Rindler horizon on an accelerated black hole. For small black holes (\(\alpha r_s/2c^2\ll 1\)) this result can be expanded to yield
$$\begin{aligned} S_{BH,R}\simeq S_{BH}^{(0)}\left( 1+\frac{S_{BH}^{(0)}}{16 S_R} + O\left[ \left( S_{BH}^0/S_R\right) ^2 \right] \right) \end{aligned}$$
(49)
with the entropy of the Rindler horizon \(S_R\) which is, of course, the result for the calculation in the asymptotic form.
Plots of the corresponding altered Hawking temperature and Bekenstein entropy are shown in Figs. 3 and 4, respectively. As stated above, the presence of a Rindler horizon decreases the temperature of a black hole thus increasing its entropy. This effect is maximal when one uses the exact formulas.
Friedmann space-time
Analogously, the entropy of a black hole surrounded by a Friedmann horizon can be obtained. Correspondingly, the Hawking temperature becomes
$$\begin{aligned} T_{H,F}=\frac{c\hbar }{k_B}\frac{1}{4\pi ^2 r_H}\left( \sqrt{\left( \frac{\pi }{2\arctan f(r_s)-\pi /2}\right) ^2-1}\right) . \end{aligned}$$
(50)
Unfortunately, the integration of the entropy cannot be done analytically. Therefore it will be given in its integral form
$$\begin{aligned} S_{BH,F}=\frac{2\pi ^2 k_B r_H}{l_p^2}\int \frac{\mathrm {d}r_s}{\sqrt{\left( \frac{\pi }{2 \arctan f(r_s)-\pi /2}\right) ^2-1}}+S_0 \end{aligned}$$
(51)
with the integration constant \(S_0,\) again, chosen in a way that \(S_{BH,F}(r_s=0)=0.\)
The expansion for small \(r_s/r_H\) reads
$$\begin{aligned} S_{BH,F}\simeq S_{BH}^{(0)}\left( 1+\frac{3+\pi ^2}{12\pi ^2}\frac{S_{BH}^{(0)}}{S_H} + O\left[ \left( S_{BH}^0/S_H \right) ^2 \right] \right) , \end{aligned}$$
(52)
where the Hubble-horizon entropy \(S_H\) equals to the asymptotic result.
Plots of the modified Hawking temperature and Bekenstein entropy are shown in Figs. 5 and 6 respectively, where the latter was computed numerically. Analogously to the Rindler case, the presence of the horizon decreases the temperature and increases the entropy. As for the Rindler horizon, the application of the exact relation results in a considerable amplification of this effect.