Thermodynamics of charged dilatonic BTZ black holes in rainbow gravity

In this paper, the charged three-dimensional Einstein's theory coupled to a dilatonic field has been considered in the rainbow gravity. The dilatonic potential has been written as the linear combination of two Liouville-type potentials. Four new classes of charged dilatonic rainbow black hole solutions, as the exact solution to the coupled field equations of the energy dependent space time, have been obtained. Two of them are correspond to the Coulomb's electric field and the others are consequences of a modified Coulomb's law. Total charge and mass as well as the entropy, temperature and electric potential of the new charged black holes have been calculated in the presence of rainbow functions. Although the thermodynamic quantities are affected by the rainbow functions, it has been found that the first law of black hole thermodynamics is still valid for all of the new black hole solutions. At the final stage, making use of the canonical ensemble method and regarding the black hole heat capacity, the thermal stability or phase transition of the new rainbow black hole solutions have been analyzed.