Erratum to: Eur. Phys. J. C (2022) 82:272 https://doi.org/10.1140/epjc/s10052-022-10232-0

In computing the quantum gravitational corrections to entropy and pressure, the wrong formula was used. The derivative of the Riemann tensor with respect to itself is

$$\begin{aligned} \frac{\partial R_{\alpha \beta \gamma \delta }}{\partial R_{\mu \nu \rho \sigma }}= & {} \frac{1}{8}\bigg (\delta ^{\mu }_{\alpha }\delta ^{\nu }_{\beta }\delta ^{\rho }_{\gamma }\delta ^{\sigma }_{\delta }-\delta ^{\mu }_{\alpha }\delta ^{\nu }_{\beta }\delta ^{\rho }_{\delta }\delta ^{\sigma }_{\gamma }\nonumber \\{} & {} -\delta ^{\mu }_{\beta }\delta ^{\nu }_{\alpha }\delta ^{\rho }_{\gamma }\delta ^{\sigma }_{\delta }+\delta ^{\mu }_{\beta }\delta ^{\nu }_{\alpha }\delta ^{\rho }_{\delta }\delta ^{\sigma }_{\gamma }+\delta ^{\mu }_{\gamma }\delta ^{\nu }_{\delta }\delta ^{\rho }_{\alpha }\delta ^{\sigma }_{\beta }\nonumber \\{} & {} -\delta ^{\mu }_{\gamma }\delta ^{\nu }_{\delta }\delta ^{\rho }_{\beta }\delta ^{\sigma }_{\alpha }-\delta ^{\mu }_{\delta }\delta ^{\nu }_{\gamma }\delta ^{\rho }_{\alpha }\delta ^{\sigma }_{\beta }+\delta ^{\mu }_{\delta }\delta ^{\nu }_{\gamma }\delta ^{\rho }_{\beta }\delta ^{\sigma }_{\alpha }\bigg ). \end{aligned}$$
(1)

Hence, (45) and (47) in [1] are correct, but (46) changes to

$$\begin{aligned} \begin{aligned} \frac{\partial ( R_{\alpha \beta }R^{\alpha \beta })}{\partial R_{\mu \nu \rho \sigma }} = \frac{1}{2}\bigg (g^{\mu \rho }R^{\nu \sigma }{-}g^{\nu \rho }R^{\mu \sigma }{-}g^{\mu \sigma }R^{\nu \rho }{+}g^{\nu \sigma }R^{\mu \rho }\bigg ). \end{aligned}\nonumber \\ \end{aligned}$$
(2)

Accordingly, the expressions of entropy and pressure are slightly modified. For the entropy:

$$\begin{aligned} S_{\text {Wald}}=\frac{A}{4G}-2\pi Q^2+S_0+S_2Q^2+{\mathcal {O}}\left( Q^4\right) , \end{aligned}$$
(3)

where

$$\begin{aligned} S_0= & {} 64\pi ^2c_3+64\pi ^2\gamma \Big [\ln \left( 4G^2_NM^2\mu ^2\right) +2\gamma _E-2\Big ], \end{aligned}$$
(4)
$$\begin{aligned} S_2= & {} \frac{4\pi ^2}{G_NM^2}\Big [5(c_2+4c_3)+\beta \left( 10\gamma _E-21\right) \nonumber \\{} & {} +8\gamma \left( 5\gamma _E-11\right) +10\left( \beta +4\gamma \right) \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +\frac{64\pi ^3}{9G^2_NM^4}\bigg \{54(\beta +4\gamma )\Big [c_1+2\alpha \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +12(12\gamma _E-23)\beta \Big [c_2+2\beta \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +48(48\gamma _E-97)\gamma \Big [c_3+2\gamma \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +6\Big [c_2\gamma (96\gamma _E-185)+c_3\beta (96\gamma _E-193)\nonumber \\{} & {} +12\beta \gamma (32\gamma _E-63)\ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +36\Big [c^2_2+4c_2\beta \ln \left( 2G_NM\mu \right) +4\beta ^2\ln ^2\left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +576\Big [c^2_3+4c_3\gamma \ln \left( 2G_NM\mu \right) +4\gamma ^2\ln ^2\left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +288\Big [c_2c_3+2c_2\gamma \ln \left( 2G_NM\mu \right) +2c_3\beta \ln \left( 2G_NM\mu \right) \nonumber \\{} & {} +4\beta \gamma \ln ^2\left( 2G_NM\mu \right) \Big ]+(\beta +4\gamma )\Big [ 9\alpha (12\gamma _E-25)\nonumber \\{} & {} +8\beta \left( 3\gamma _E(6\gamma _E-23)+40+3\pi ^2\right) \nonumber \\{} & {} +4\gamma \big (6\gamma _E\left( 24\gamma _E-97\right) +331+30\pi ^2\big )\Big ] \bigg \}. \end{aligned}$$
(5)

For the pressure:

$$\begin{aligned} P=-\frac{Q^2}{64\pi G^4_NM^4}-\frac{\gamma }{2G^4_N M^4}+P_2Q^2+{\mathcal {O}}\left( Q^4\right) \end{aligned}$$
(6)

where

$$\begin{aligned} P_2= & {} \frac{1}{32G^5_NM^6}\Big [c_2+4c_3+2\beta (\gamma _E-4)\nonumber \\{} & {} +8\gamma (\gamma _E-5)+2(\beta +4\gamma )\ln (2G_NM\mu )\Big ]\nonumber \\{} & {} +\frac{\pi }{9G^6_NM^8}\bigg \{54(\beta +4\gamma )\Big [c_1+2\alpha \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +24(6\gamma _E-13)\beta \Big [c_2+2\beta \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +768(3\gamma _E-7)\gamma \Big [c_3+2\gamma \ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +6\Big [c_2\gamma (96\gamma _E-215)+c_3\beta (96\gamma _E-217)\nonumber \\{} & {} +96\beta \gamma (4\gamma _E-9)\ln \left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +36\Big [c^2_2+4c_2\beta \ln \left( 2G_NM\mu \right) +4\beta ^2\ln ^2\left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +576\Big [c^2_3+4c_3\gamma \ln \left( 2G_NM\mu \right) +4\gamma ^2\ln ^2\left( 2G_NM\mu \right) \Big ]\nonumber \\{} & {} +288\Big [c_2c_3+2c_2\gamma \ln \left( 2G_NM\mu \right) +2c_3\beta \ln \left( 2G_NM\mu \right) \nonumber \\{} & {} +4\beta \gamma \ln ^2\left( 2G_NM\mu \right) \Big ]+(\beta +4\gamma )\Big [36\alpha (3\gamma _E-7)\nonumber \\{} & {} +2\beta \big (8\gamma _E(9\gamma _E-39)+229+12\pi ^2\big )\nonumber \\{} & {} +\gamma \big (192\gamma _E(3\gamma _E-14)+2095+120\pi ^2\big )\Big ]\bigg \}. \end{aligned}$$
(7)