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Criticality and extended phase space thermodynamics of AdS black holes in higher curvature massive gravity

Considering de Rham-Gabadadze-Tolley theory of massive gravity coupled with (ghost free) higher curvature terms arisen from the Lovelock Lagrangian, we obtain charged AdS black hole solutions in diverse dimensions. We compute thermodynamic quantities in the extended phase space by considering the variations of the negative cosmological constant, Lovelock coefficients ($α_{i}$) and massive couplings ($c_{i}$), and prove that such variations is necessary for satisfying the extended first law of thermodynamics as well as associated Smarr formula. In addition, by performing a comprehensive thermal stability analysis for the topological black hole solutions, we show in what regions thermally stable phases exist. Calculations show the results are radically different from those in Einstein gravity. Furthermore, we investigate $P-V$ criticality of massive charged AdS black holes in higher dimensions, including the effect of higher curvature terms and massive parameter, and find that the critical behavior and phase transition can happen for non-compact black holes as well as spherically symmetric ones. The phase structure and critical behavior of topological AdS black holes are drastically restricted by the geometry of event horizon. In this regard, the universal ratio, i.e. $\frac{{{P_c}{v_c}}}{T_c}$, is a function of the event horizon topology. It is shown the phase structure of AdS black holes with non-compact (hyperbolic) horizon could give birth to three critical points corresponds to a reverse van der Waals behavior for phase transition which is accompanied with two distinct van der Waals phase transitions. For black holes with spherical horizon, the van der Waals, reentrant and analogue of solid/liquid/gas phase transitions are observed.