What does the discovery of pressure of black holes imply for spacetime thermodynamics? As far as I know AdS/CFT and spacetime thermodynamics was motivated by the Bekenstein-Hawking Entropy formula of black holes.
Recently however it was discovered that black holes have a pressure as well.
Does this have any implications for spacetime thermodynamics?
 A: The pressure of Schwarzschild black holes arises as a consequence of the quantum corrections due to local and non-local terms in the effective quantum gravity action.
For a Schwarzschild black hole it was found in $[1]$ that
$$P_{q,S} = − \gamma \frac{1}{2 G^4_N M^4}$$
where $\gamma$ is a constant appearing in the non-local part of the action.
However, charged (Reissner-Nordström) black holes do possess a classical pressure $[2]$, which is proportional to the charge, and hence is zero for Schwarzschild:
$$ P_{cl,RN}=-\frac{Q^2}{64\pi G^4_NM^4} $$
Quantum gravity effects modify this classical expression too. The full expression for the pressure was found in $[3]$ (with some approximations) and is
$$ P_{RN}=P_{cl,RN}+P_{q,S}+P_{q,RN}Q^2$$
where $P_{q,RN}$ is the a slightly more complicated generalisation of $P_{q,S}$:
\begin{multline}
    P_{q,RN}=\frac{1}{32G^5_NM^6}\Big[c_2+4c_3+2\beta(\gamma_E-4)+8\gamma(\gamma_E-5)+2(\beta+4\gamma)\ln(2G_NM\mu)\Big]
    \\+\frac{\pi}{9G^6_NM^8}\bigg\{54(\beta+4\gamma)\Big[c_1+2\alpha\ln\left(2G_NM\mu\right)\Big]
    +72(3\gamma_E-7)\beta\Big[c_2+2\beta\ln\left(2G_NM\mu\right)\Big]\\
    +768(3\gamma_E-7)\gamma\Big[c_3+2\gamma\ln\left(2G_NM\mu\right)\Big]
    +6\Big[c_2\gamma(120\gamma_E-287)+3c_3\beta(40\gamma_E-91)\\
    +160\beta\gamma(3\gamma_E-7)\ln\left(2G_NM\mu\right)\Big]
     +54\Big[c^2_2+4c_2\beta\ln\left(2G_NM\mu\right)+4\beta^2\ln^2\left(2G_NM\mu\right)\Big]\\
   +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]
   +360\Big[c_2c_3+2c_2\gamma\ln\left(2G_NM\mu\right)\\+2c_3\beta\ln\left(2G_NM\mu\right)
    +4\beta\gamma\ln^2\left(2G_NM\mu\right)\Big]+(\beta+4\gamma)\Big[36\alpha(3\gamma_E-7) \\+9\beta\big(8\gamma_E(3\gamma_E-14)+95+4\pi^2\big)
    +\gamma\big(192\gamma_E(3\gamma_E-14)+2095+120\pi^2\big)\Big]  
    \bigg\}.
    \end{multline}
Here, $\alpha,\beta,\gamma$ belong to the non-local part of the action, $c_1,c_2,c_3$ belong to the local part.
The quantum corrections do not only modify the pressure, but also the entropy, the temperature etc. The effective field theory method for quantum gravity is useful because you can keep track of the quantum effects easily.
References
$[1]$ X. Calmet and F. Kuipers. Quantum gravitational corrections to the entropy of a Schwarzschild black hole. Phys. Rev. D 104 , 066012 (2021).
$[2]$Y.-H. Wei. Understanding first law of thermodynamics of black holes. Phys. Lett. B 672 98–100 (2009).
$[3]$ Campos Delgado, R. Quantum gravitational corrections to the entropy of a Reissner–Nordström black hole. Eur. Phys. J. C 82, 272 (2022).
