Quantum corrected thermodynamics of nonlinearly charged BTZ black holes in massive gravity's rainbow

In this paper, we consider three-dimensional massive gravity's rainbow and obtain black hole solutions in three different cases of Born-Infeld, logarithmic, and exponential theories of nonlinear electrodynamics. We discuss the horizon structure and geometrical properties. Then, we study thermodynamics of these models by considering the first-order quantum correction effects, which appear as a logarithmic term in the black hole entropy. We discuss such effects on the black hole stability and phase transitions. We find that due to the quantum corrections, the second-order phase transition happens in Born-Infeld and logarithmic models. We obtain the modified first law of black hole thermodynamics in the presence of logarithmic corrections.