Thermodynamics, phase structure of Bardeen massive black hole in Gauss-Bonnet gravity

This work provides the exact solution of the Bardeen black hole in association with 4D Gauss-Bonnet massive gravity in Anti-de-Sitter (AdS) space-time. It is a modification of the Gauss-Bonnet when gravity couples with nonlinear matter fields which is the function of the electromagnetic field. The obtained solution gives rise to 4D EGB Bardeen black holes when the massive gravity parameter is set to zero and it yields a 4D Gauss-Bonnet black hole in the absence of magnetic monopole charge. Further, we analyze and adopt the thermodynamic quantities like mass (M+), temperature (T+) and heat capacity (C+) in the presence of massive gravity and nonlinear electrodynamics. In addition, we extend our results by considering the cosmological constant (?) as a thermodynamical variable (P = -?/8p) and obtain the critical values of pressure, temperature, horizon radius and analyze the behavior of the global parameter Pcvc/Tc. The effect of a massive parameter (m) of the critical exponent is opposite to the magnetic monopole charge (e) and Gauss-Bonnet parameter (a). According to our analysis the phase transition between a small and large black hole and van der Waals phase transition are analogous to each other.