Thermodynamic Instabilities of Generalized Exotic BTZ Black Holes

We examine the conjecture that black holes violating the reverse isoperimetric inequality have negative specific heat at constant volume $C_V$. We test this conjecture on the family of generalized exotic Ba\~nados, Teitelbiom and Zanelli (BTZ) black holes and find that $C_V$ can be positive even when the reverse isoperimetric inequality is violated, providing a counter example to the conjecture. However in all cases where $C_V$ is positive, the specific heat at constant pressure $C_P$ is negative, indicating that generalized exotic black holes are thermodynamically unstable, suggesting that a broader version of the conjecture might hold.