Extended phase space of AdS black holes in Einstein-Gauss-Bonnet gravity with a quadratic nonlinear electrodynamics

In this paper, we consider quadratic Maxwell invariant as a correction to the Maxwell theory and study thermodynamic behavior of the black holes in Einstein and Gauss-Bonnet gravities. We consider cosmological constant as a thermodynamic pressure to extend phase space. Next, we obtain critical values in case of variation of nonlinearity and Gauss-Bonnet parameters. Although the general thermodynamical behavior of the black hole solutions is the same as usual Van der Waals system, we show that in special case of the nonlinear electromagnetic field, there will be a turning point for the phase diagrams and usual Van der Waals is not observed. This theory of nonlinear electromagnetic field provides two critical horizon radii. We show that this unusual behavior of phase diagrams is due to existence of second critical horizon radius. It will be pointed out that the power of the gravity and nonlinearity of the matter field modify the critical values. We generalize the study by considering the effects of dimensionality on critical values and make comparisons between our models with their special sub-classes. In addition, we examine the possibility of the existence of the reentrant phase transitions through two different methods.