# Extended Thermodynamics and Complexity in Gravitational Chern-Simons Theory

## TL;DR

This paper explores the extended thermodynamics of BTZ black holes with generalized mass and angular momentum parameters, revealing distinct behaviors, thermodynamic properties, and complexity growth patterns, including the special case of purely exotic black holes.

## Contribution

It introduces a generalized framework for BTZ black hole thermodynamics with arbitrary parameters, analyzing their properties, stability, and complexity growth, including exotic cases.

## Key findings

- Existence of two distinct thermodynamic behaviors based on parameters
- Thermodynamic volumes are positive under certain constraints
- Complexity growth is proportional to that of standard BTZ black holes

## Abstract

We study several aspects of the extended thermodynamics of BTZ black holes with thermodynamic mass $M=\alpha m + \gamma \frac{j}{\ell}$ and angular momentum $J = \alpha j + \gamma \ell m$, for general values of the parameters $(\alpha, \gamma)$ ranging from regular ($\alpha=1, \gamma=0$) to exotic ($\alpha=0, \gamma=1$). We show that there exist two distinct behaviours for the black holes, one when $\alpha > \gamma$ ("mostly regular"), and the other when $\gamma < \alpha$ ("mostly exotic"). We find that the Smarr formula holds for all $(\alpha, \gamma)$. We derive the corresponding thermodynamic volumes, which we find to be positive provided $\alpha$ and $\gamma$ satisfy a certain constraint. The dependence of pressure on volume is unremarkable and strictly decreasing when $\alpha > \gamma$, but a maximum volume emerges for large $J\gg T$ when $\gamma > \alpha$; consequently an exotic black hole of a given horizon circumference and temperature can exist in two distinct anti de Sitter backgrounds. We compute the reverse isoperimetric ratio, and study the Gibbs free energy and criticality conditions for each. Finally we investigate the complexity growth of these objects and find that they are all proportional to the complexity of the BTZ black hole. Somewhat surprisingly, purely exotic BTZ black holes have vanishing complexity growth.