# Thermodynamic Instabilities of Generalized Exotic BTZ Black Holes

## TL;DR

This paper investigates the thermodynamic stability of generalized exotic BTZ black holes, challenging a previous conjecture by showing counterexamples where specific heat conditions do not align with the inequality.

## Contribution

It provides the first counterexamples to the conjecture linking reverse isoperimetric inequality violation with negative specific heat in black holes.

## Key findings

- Counterexamples where $C_V$ is positive despite inequality violation
- All cases with positive $C_V$ have negative $C_P$, indicating instability
- Broader conjecture may be necessary to describe black hole thermodynamics

## Abstract

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.