# Topological black hole in the theory with nonminimal derivative coupling with power-law Maxwell field and its thermodynamics

## TL;DR

This paper derives topological black hole solutions in scalar-tensor gravity with nonminimal derivative coupling and power-law Maxwell field, analyzing their thermodynamics, stability, and phase transitions.

## Contribution

It introduces new black hole solutions with nonminimal derivative coupling and power-law Maxwell field, extending previous models and exploring their thermodynamic properties.

## Key findings

- Black hole solutions generalize previous charged solutions.
- Thermodynamic analysis reveals stability and phase transition behavior.
- Heat capacity varies significantly with parameters, indicating diverse thermodynamic phases.

## Abstract

We obtain topological black hole solutions in scalar-tensor gravity with nonminimal derivative coupling between scalar and tensor components of gravity and power-law Maxwell field minimally coupled to gravity. The obtained solutions can be treated as a generalization of previously derived charged solutions with standard Maxwell action \cite{Feng_PRD16}. We examine the behaviour of obtained metric functions for some asymptotic values of distance and coupling. To obtain information about singularities of the metrics we calculate Kretschmann scalar. We also examine the behaviour of gauge potential and show that it is necessary to impose some constraints on parameter of nonlinearity in order to obtain reasonable behaviour of the filed. The next part of our work is devoted to the examination of black hole's thermodynamics. Namely we obtain black hole's temperature and investigate it in general as well as in some particular cases. To introduce entropy we use well known Wald procedure which can be applied to quite general diffeomorphism-invariant theories. We also extend thermodynamic phase space by introducing thermodynamic pressure related to cosmological constant and as a result we derive generalized first law and Smarr relation. The extended thermodynamic variables also allow us to construct Gibbs free energy and its examination gives important information about thermodynamic stability and phase transitions. We also calculate heat capacity of the black holes which demonstrates variety of behaviour for different values of allowed parameters.