Dilatonic black holes in gravity's rainbow with a nonlinear source: the effects of thermal fluctuations

This paper is devoted to investigate nonlinearly charged dilatonic black holes in the context of gravity's rainbow with two cases: I) by considering the usual entropy, II) in the presence of first order logarithmic correction of entropy. First, exact solutions of dilatonic Born-Infeld gravity with an energy dependent Liouville-type potential with black hole interpretation are obtained. Then, thermodynamic properties of the mentioned cases are studied, separately. It will be shown that although mass, entropy and heat capacity are modified due to the presence of first order correction, the temperature remains independent of it. Furthermore, it will be shown that divergencies of the heat capacity, hence phase transition points are also independent of first order correction whereas the stability conditions are highly sensitive to variation of the correction parameter. Except for the effects of first order correction, we will also draw a limit on values of the dilatonic parameter and show that it is possible to recognize AdS and dS thermodynamical behaviors for two specific branches of the dilatonic parameter. In addition, the effects of nonlinear electromagnetic field and energy functions on thermodynamical behavior of the solutions will be highlighted and dependency of critical behavior, on these generalizations will be investigated.