Given the blackhole temperature formula, $$T =frac{hbar c^3}{8pi k_BGM},$$ can I use $M=E/c^2$ (where $M$ is the mass of the Blackhole) to invert the above equation to express it as $$E=E(T)$$ and use the formula $C_V=left(frac{partial E}{partial T}right)_{V,P}$ to calculate the specific heat of the Blackhole? This little piece of calculation leads to a negative specific heat $$C_V=-frac{hbar c^5}{8pi Gk_BT^2}.$$ Will this also be true for a rotating black hole and a charged black hole because for both of which I suspect $Eneq Mc^2$?

Thermodynamics from the total energy of a Black hole and its temperature

Physics Asked on December 12, 2020

Given the blackhole temperature formula, $$T =frac{hbar c^3}{8pi k_BGM},$$ can I use $M=E/c^2$ (where $M$ is the mass of the Blackhole) to invert the above equation to express it as $$E=E(T)$$ and use the formula $C_V=left(frac{partial E}{partial T}right)_{V,P}$ to calculate the specific heat of the Blackhole? This little piece of calculation leads to a negative specific heat $$C_V=-frac{hbar c^5}{8pi Gk_BT^2}.$$
Will this also be true for a rotating black hole and a charged black hole because for both of which I suspect $Eneq Mc^2$?

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