Charged Dilatonic Black Holes in Gravity's Rainbow

In this paper, we present charged dilatonic black holes in gravity's rainbow. We study geometric and thermodynamic properties of black hole solutions. We also investigate the effects of rainbow functions on different thermodynamic quantities for these charged black holes in dilatonic gravity's rainbow. Then, we demonstrate that first law of thermodynamics is valid for these solutions. After that, we investigate thermal stability of the solutions using canonical ensemble and analyze the effects of different rainbow functions on thermal stability. In addition, we present some arguments regarding the bound and phase transition points in context of geometrical thermodynamics. We also study the phase transition in extended phase space in which cosmological constant is treated as the thermodynamic pressure. Finally, we use another approach to calculate and demonstrate that obtained critical points in extended phase space are representing a second order phase transition for these black holes.