Thermodynamics of charged four-dimensional hairy black holes

In this paper, we study the solutions of Einstein’s field equations in Einstein–Maxwell-dilaton gravity. By presenting the (3+1)-dimensional action in which gravity is coupled to a dilaton field, we derive the field equations. We present a new class of charged dilaton black hole solutions in Einstein–Maxwell-dilaton theory and investigate the effects of dilaton field on the properties of space–time. We show that these solutions are physical and describe spherically symmetric charged black holes. The calculated potential is Liouville type and has two rules. The exact solution of the field equations leads to the production of the metric function with three rules. By examining the physical properties of the solutions, we find the mass, charge, electric potential and temperature of charged dilaton black holes. Also, with the help of the canonical ensemble method, we study the thermodynamics and thermal stability of solutions and show the effects of dilaton and Maxwell fields on the thermodynamics of these solutions. Finally, by examining the phase transition of the solutions, we present thermal stability of the black hole in (3+1)-dimensional.