# Thermodynamical stability of $f(R)$-AdS black holes in grand canonical ensemble

## TL;DR

This paper investigates the thermodynamical stability and phase transitions of $f(R)$-AdS black holes in higher dimensions, revealing different transition types and the effectiveness of thermodynamical geometry methods near specific phase transition points.

## Contribution

It classifies $f(R)$-AdS black holes into SAdS-like and RN-AdS-like, analyzes their stability and phase transitions, and assesses the accuracy of thermodynamical geometry methods in grand canonical ensemble.

## Key findings

- RN-AdS-like black holes undergo two types of phase transitions.
- SAdS-like black holes experience only one type of phase transition.
- Thermodynamical extrinsic curvature accurately indicates stability near type-two transitions.

## Abstract

In this paper, $n$-dimensional ($n=4p, p$ is a positive integer) $f(R)$-AdS black holes is divided into Schwazschild-AdS (SAdS) like ones and Reissner-Nordstr\"{o}m-AdS (RN-AdS) like ones. Thermodynamical stability of them in grand canonical ensemble is investigated. Locally, we find that the RN-AdS like $f(R)$ black holes will experience either type-one or type-two phase transitions from unstable states to stable states, whereas there are only type-two phase transitions between that two states for SAdS like $f(R)$ black holes. Globally, we find that there are type-one Hawking-Page like phase transitions between thermal AdS state and $f(R)$ black holes. Using thermodynamical geometry method, we find that thermodynamical extrinsic curvature can only provide accurate stability information near type-two (not type-one) phase transition points for $f(R)$ black holes in grand canonical ensemble.