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We investigate a new class of $(n+1)$-dimensional topological black holesolutions in the context of massive gravity and in the presence of logarithmicnonlinear electrodynamics. Exploring higher dimensional solutions in massivegravity coupled to nonlinear electrodynamics is motivated by holographichypothesis as well as string theory. We first construct exact solutions of thefield equations and then explore the behavior of the metric functions fordifferent values of the model parameters. We observe that our black holes admitthe multi-horizons caused by a quantum effect called anti-evaporation. Next, bycalculating the conserved and thermodynamic quantities, we obtain a generalizedSmarr formula. We find that the first law of black holes thermodynamics issatisfied on the black hole horizon. We study thermal stability of the obtainedsolutions in both canonical and grand canonical ensembles. We reveal thatdepending on the model parameters, our solutions exhibit a rich variety ofphase structures. Finally, we explore, for the first time without extendingthermodynamics phase space, the critical behavior and reentrant phasetransition for black hole solutions in massive gravity theory. We realize thatthere is a zeroth order phase transition for a specified range of charge valueand the system experiences a large/small/large reentrant phase transition dueto the presence of nonlinear electrodynamics.

Thermodynamics and reentrant phase transition for logarithmic nonlinear charged black holes in massive gravity