Instability of Super-Entropic Black Holes in Extended Thermodynamics

The charged black hole of Banados, Teitelbiom and Zanelli is studied in extended gravitational thermodynamics where there is a dynamical pressure and volume. It is a simple example of a super-entropic black hole, violating the reverse isoperimetric inequality. It is proven that this property implies that its specific heat at constant volume is negative, signalling a new kind of fundamental instability for black holes. It is conjectured that this instability is present for other super-entropic black holes, and this is demonstrated numerically for a large family of known solutions.