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

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.