P-v criticality, phase structure and extended thermodynamics of AdS ABG black holes

This paper presents the Ayon-Beato Garcia black hole (ABG) in AdS spacetime known as regular black hole solutions at origin ( r→ 0 ); it has anti-de Sitter and de Sitter form depending upon the charge-to-mass ratio >1 and <1 , respectively. To investigate the solution, we compute the thermodynamic quantities mass, temperature, entropy and specific heat in extended phase space. The cosmological constant Λ is expressed in terms of thermodynamical pressure P as P=-Λ /8π . By observing the behaviour of the Gibbs free energy, we study the first- and second-order phase transition. The first-order phase transition is seen at below the critical point, and it terminates at the critical point where the phase transition becomes second order which is characterized by the critical exponents. We also explore the equation of state for black holes and their critical exponents and prove it to be the same those of Van der Waals fluid. In the case of regular black hole, the area law is violated. For the ABG black hole, a new thermodynamics quantity ϕ _M is defined which is required for the consistency of both first law of black hole thermodynamics and Smarr relation.