Exponential Modification of $4D$ AdS Black Hole and Thermodynamic Behavior

In this paper, we present an exponential modification for the action of an AdS black hole in the absence of a matter field. An approximated black hole solution is obtained up to the third order of perturbation coefficient. A thermodynamic investigation in canonical ensemble shows that the behavior of a Van der Waals fluid is not seen in this model. Nevertheless, the study of thermodynamic potentials and other related quantities suggests that the thermodynamic phase transitions of the first and second types can occur in this model. The forms of the phase transitions are more similar to the Hawking-Page phase transitions.