Thermodynamics of novel charged dilaton black holes in gravity's rainbow

In the present work, we have studied the thermodynamical properties of black holes arising as the solutions of the four-dimensional dilaton gravity coupled to Maxwell's electrodynamics in gravity's rainbow. This theory allows three classes of asymptotically non-flat and non-AdS black hole solutions. We showed that the self-interacting scalar function, as the solution to the scalar field equation, can be written as the linear combination of three Liouville-type potentials. The thermodynamical quantities are identified and in particular, a generalized Smarr formula is derived. It is shown that, although the thermodynamic quantities are affected by the rainbow functions, the validity of the black hole thermodynamical first law is supported. The thermodynamic stability of the solutions have been analyzed through the black hole heat capacity. We have shown that, even in the presence of the rainbow functions, the black holes can be locally stable in the sense that there exists a range of the horizon radiuses for which the heat capacity is positive.