Thermodynamic instability of nonlinearly charged black holes in gravity's rainbow

Motivated by the violation of Lorentz invariancy in quantum gravity, we study black hole solutions in gravity's rainbow in context of Einstein gravity coupled with various models of nonlinear electrodynamics. We regard an energy dependent spacetime and obtain related metric functions and electric fields. We show that there is an essential singularity at the origin which is covered with an event horizon. We also compute the conserved and thermodynamical quantities and examine the validity of the first law of thermodynamics in the presence of rainbow functions. Finally, we investigate thermal stability conditions for these black hole solutions in context of canonical ensemble. We show that thermodynamical structure of the solutions depends on the choices of nonlinearity parameters, charge and energy functions.