  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.
