  We investigate a new class of $(n+1)$-dimensional topological black hole
solutions in the context of massive gravity and in the presence of logarithmic
nonlinear electrodynamics. Exploring higher dimensional solutions in massive
gravity coupled to nonlinear electrodynamics is motivated by holographic
hypothesis as well as string theory. We first construct exact solutions of the
field equations and then explore the behavior of the metric functions for
different values of the model parameters. We observe that our black holes admit
the multi-horizons caused by a quantum effect called anti-evaporation. Next, by
calculating the conserved and thermodynamic quantities, we obtain a generalized
Smarr formula. We find that the first law of black holes thermodynamics is
satisfied on the black hole horizon. We study thermal stability of the obtained
solutions in both canonical and grand canonical ensembles. We reveal that
depending on the model parameters, our solutions exhibit a rich variety of
phase structures. Finally, we explore, for the first time without extending
thermodynamics phase space, the critical behavior and reentrant phase
transition for black hole solutions in massive gravity theory. We realize that
there is a zeroth order phase transition for a specified range of charge value
and the system experiences a large/small/large reentrant phase transition due
to the presence of nonlinear electrodynamics.
