Specific Heats for Quantum BTZ Black Holes in Extended Thermodynamics

It was shown recently that extended black hole thermodynamics, where the cosmological constant is a dynamical variable, giving rise to a pressure $p$ and its conjugate volume $V$, can be given a natural setting in the context of braneworld models. We study the specific heat capacities $C_p(T)$ and $C_V(T)$ of the quantum version of the BTZ black hole that lives in the induced gravity theory on the brane. There are multiple branches of solutions, and we explore and characterize key features of the possible behaviour. We identify and study a critical point in the space of solutions where both specific heats diverge. In the regime of weak backreaction where we a close to an ordinary theory of gravity, the black hole is "sub-entropic", but as backreaction is increased we note that there are parts of parameter space that has regions where it is "super-entropic". While a study of the sign of the specific heats does not always show a corresponding instability (conjectured in the literature), the presence of strong backreaction makes interpretation unclear.