Critical behaviors and phase transitions of black holes in higher order gravities and extended phase spaces

We consider the critical behaviors and phase transitions of Gauss Bonnet-Born Infeld-AdS black holes (GB-BI-AdS) for $d=5,6$ and the extended phase space. We assume the cosmological constant, $Lambda$, the coupling coefficient $alpha$, and the BI parameter $beta$ to be thermodynamic pressures of the system. Having made these assumptions, the critical behaviors are then studied in the two canonical and grand canonical ensembles. We find reentrant and triple point phase transitions (RPT-TP) and multiple reentrant phase transitions (multiple RPT) with increasing pressure of the system for specific values of the coupling coefficient $alpha$ in the canonical ensemble. Also, we observe a reentrant phase transition (RPT) of GB-BI-AdS black holes in the grand canonical ensemble and for $d=6$. These calculations are then expanded to the critical behavior of Born-Infeld-AdS (BI-AdS) black holes in the third order of Lovelock gravity and in the grand canonical ensemble to find a Van der Waals behavior for $d=7$ and a reentrant phase transition for $d=8$ for specific values of potential $phi$ in the grand canonical ensemble. Furthermore, we obtain a similar behavior for the limit of $beta to infty$, i.e charged-AdS black holes in the third order of the Lovelock gravity. Thus, it is shown that the critical behaviors of these black holes are independent of the parameter $beta$ in the grand canonical ensemble.