Three dimensional dilatonic gravity's rainbow: exact solutions

Deep relations of dark energy scenario and string theory results into dilaton gravity, on one hand, and the connection between quantum gravity with gravity's rainbow, on the other hand, motivate us to consider three dimensional dilatonic black hole solutions in gravity's rainbow. We obtain two classes of the solutions which are polynomial and logarithmic forms. We also calculate conserved and thermodynamic quantities, and examine the first law of thermodynamics for both classes. In addition, we study thermal stability and show that one of the classes is thermally stable while the other one is unstable.