Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / c / ii / Solution

For $a=0$, $T_H=(8\pi M{)}^{-1}$, so the Schwarzschild black hole has negative heat capacity. A small energy gain from a reservoir lowers its temperature below the reservoir temperature and causes further absorption; a small energy loss raises its temperature and causes further emission. The equilibrium is therefore unstable in the canonical ensemble.