  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.
