  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.
