Quantum pressure and chemical potential for a Schwarzschild black hole? Just as Hawking showed that even Schwarzschild black holes have a temperature, shouldn't they also have a pressure and chemical potential? Are there any analytical formulae of those as well as
$$ T_{BH}=\dfrac{\hbar c^3}{8\pi GMk_B}$$?
I guess for the pressure (energy density) of Schwarzschild black holes this quantity:
$$P_S=\dfrac{E_S}{V_S}=\dfrac{k_BT_{BH}}{\dfrac{4\pi}{3}\left(\dfrac{2GM}{c^2}\right)^3}=\dfrac{3\hbar c^9}{256\pi^2G^4M_\odot^4}\left(\dfrac{M_\odot}{M}\right)^4 \approx 8\cdot 10^{-42}\left(\dfrac{M_\odot}{M}\right)^4\;\; Pa$$
However I am not sure if it is meaningful and how to guess the chemical potential (if any) for (quantum) Schwarzschild black holes... Has it any sense or not?
 A: Yes, in spacetime dynamics, and specifically for a black hole, there is a corresponding pressure, energy density, entropy, and even free energy of which can be calculated. A quick introduction (without the details such as those from Hawking's original paper "4 Laws of Black Hole Thermodynamics") is in Natsuume's 'user guide' on AdS/CFT, chapter 3:"AdS/CFT Dualiy Guide".
So, using that reference, you can calculate the energy using the usual rules equations of thermodynamics, such as (eq. 3.33) $E = F + TS = \frac{r_0}{2G} = M$. Now, for chemical potential, think first about the original/classical meaning of $\mu$: how the different types (or species) of particles are emitted or absorbed with respect to the change in particle number. Now, recall how the Schwarzschild solution comes about: we are in a vacuum (no stress-energy tensor)! So our $\mu = 0$. But, for a charged black hole, the chemical potential is derived as (eq. 3.59, Natsuume), $$ \mu = A_0|_{r = \infty} - A_0|_{r=r_+} = \frac{Q}{r_+} = \sqrt{\frac{r_-}{Gr_+}}.$$
As a final note, these relationships are true regardless of AdS/CFT, so ignore that part of the guide unless you want to dive into it. Have fun.
A: A recent paper identifies the pressure of Schwarzschild black holes with the quantum corrections due to nonlocal terms in the effective quantum gravity action. In particular for a black hole of mass $M$ they find
$$P = − \gamma \frac{1}{2 G^4_N M^4}$$
where $\gamma$ is a positive constant for fields of spin 0, 1/2 and 2, and negative for spin 1.
A: Black holes have pressure due to anomalous behavior of gravity under the influence of quantum - gravitational effect. This pressure is mathematically equal to $P = \frac{hc^9}{G^4m^4}$ its detail can be found in Ref1
[1]https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4288217
