Non-linear charged AdS black hole in massive gravity

In the context of the non-linear electrodynamics and the Einstein-massive gravity, we have obtained a $$4\hbox {D}$$ non-linear charged AdS black hole solution. Then, we investigated its horizon structure. In addition, the thermodynamics and phase structure of this black hole solution have been studied in details. We have computed various thermodynamic quantities of the black hole, such as the temperature, entropy, the heat capacity at constant pressure, or the Gibbs free energy. The black hole can undergo the first-order, second-order phase transitions which depend crucially on the effective horizon curvature, the sign of the coupling parameter $$c_1$$ , the characteristic parameter of the non-linear electrodynamics, as well as the pressure. Finally, we derived the equation of state and studied $$P-V$$ criticality in the case of the positive effective horizon curvature.

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