The RN metric in asymptotically flat spacetime is given here: Wikipedia page on RN metric. From the metric, we can work out the Bekenstein-Hawking entropy to be S = Area/4 , which explicitly would be: S = $\pi (M+\sqrt{M^2-Q^2})^2$, where we choose the units appropriately($\hbar = G_{N} = 1$).
The temperature can be easily computed using the periodicity trick, and it comes out to be:
T= $\frac{\sqrt{M^2-Q^2}}{2\pi (M+\sqrt{M^2-Q^2})^2 }$
I want to compute the heat capacity of the RN black hole, which should be given by $\partial Q/\partial T$ at a fixed volume. From my computation, I find that the heat capacity as follows:
$ C_{V} = \frac{2\pi \sqrt{M^2-Q^2} (M+\sqrt{M^2-Q^2})^2}{M-2\sqrt{M^2-Q^2}}$
As can be observed, it can be either positive or negative, depending on the parameters M and Q. This is unlike the Schwarzhcild Black hole in asymptotically flat spacetimes, which has -ve heat capacity.
I want to understand this difference physically. Why does the addition of charges slowly (s.t Q approaches M) change the $C_{V}$ from -ve to +ve?
