Abstract
Through exact solution of the coupled scalar and gravitational field equations, in an energy dependent spacetime, two classes of novel dilatonic BTZ black holes have been found. The black hole solutions have only one horizon and their asymptotic behaviors are non-flat and non-AdS. The black hole mass, entropy and temperature have been calculated, as the conserved and thermodynamic quantities, and it has been shown that, although these quantities get modified in the presence of rainbow functions, they satisfy the first law of black hole thermodynamics in its standard form. The black hole heat capacity and Gibbs free energy have been calculated and the local and global stabilities of the black holes have been analyzed making use of the canonical and grand canonical ensembles, respectively. Then, by considering the black hole thermal fluctuations, the quantum gravitational effects on the local and global stabilities have been studied.
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