  Employing higher curvature corrections to Einstein--Maxwell gravity has
garnered a great deal of attention motivated by the high energy regime in
quantum nature of black hole physics. In addition, one may employ gravity's
rainbow to encode quantum gravity effects into the black hole solutions. In
this paper, we regard an energy dependent static spacetime with various
topologies and study its black hole solutions in the context of Gauss--Bonnet
Born--Infeld (GB--BI) gravity. We study thermodynamic properties and examine
the first law of thermodynamics. Using suitable local transformation, we endow
the Ricci--flat black hole solutions with a global rotation and study the
effects of rotation on thermodynamic quantities. We also investigate thermal
stability in canonical ensemble through calculating the heat capacity. We
obtain the effects of various parameters on the horizon radius of stable black
holes. Finally, we discuss second order phase transition in the extended phase
space thermodynamics and investigate the critical behavior.
