Inspired by applications of gravitys rainbow in UV completion of general relativity, we investigate charged topological black holes in gravitys rainbow and show that depending on the values of different parameters, these solutions may encounter with black hole solutions with two horizons, extreme black hole (one horizon) or naked singularity (without horizon). First, we obtain black hole solutions, calculate thermodynamical quantities of the system and check the first law of thermodynamics. Then, we study the thermodynamical behavior of the system including thermal stability and phase transitions. In addition, we employ geometrical thermodynamics to probe phase transition points and limits on having physical solutions. Finally, we obtain heat engines corresponding to these black holes. The goal is to see how black holes parameters such as topological factor and rainbow functions would affect efficiency of the heat engines.
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