| Summary: | Two new families of nonlinearly charged and asymptotically anti-de Sitter (AdS) rainbow black holes have been introduced, as the exact solutions to the coupled electromagnetic and gravitational field equations, in massive gravity theory and in the presence of power-law nonlinear electrodynamics. The conserved and thermodynamic quantities such as black hole mass, charge, temperature, entropy and electric potential have been calculated from geometric and thermodynamic approaches. Despite the fact that some of them receive corrections from rainbow functions and nonlinear electrodynamics, it has been proved that they fulfill the first law of black hole thermodynamics. Thermal stability of the black holes has been studied by use of the canonical ensemble and geometrical thermodynamics methods, separately. Regarding the black hole heat capacity and thermodynamic Ricci scalars, the points of type-one and type-two phase transitions and the conditions under which the black holes remain locally stable have been determined. Also, global stability of the novel rainbow black holes has been investigated from the viewpoint of the grand canonical ensemble. Through calculation of Gibbs free energy of the black holes, the points of Hawking-Page phase transition and the size of the black holes which are globally stable or in the radiative phase have been determined. Keywords: Three-dimensional black hole, Nonlinear electrodynamics, Massive gravity theory, Gravity's rainbow
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