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We consider the critical behaviors and phase transitions of Gauss Bonnet-BornInfeld-AdS black holes (GB-BI-AdS) for $d=5,6$ and the extended phase space. Weassume 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 thetwo canonical and grand canonical ensembles. We find "reentrant and triplepoint phase transitions" (RPT-TP) and "multiple reentrant phase transitions"(multiple RPT) with increasing pressure of the system for specific values ofthe coupling coefficient $\alpha$ in the canonical ensemble. Also, we observe areentrant phase transition (RPT) of GB-BI-AdS black holes in the grandcanonical ensemble and for $d=6$. These calculations are then expanded to thecritical behavior of Born-Infeld-AdS (BI-AdS) black holes in the third order ofLovelock gravity and in the grand canonical ensemble to find a Van der Waalsbehavior for $d=7$ and a reentrant phase transition for $d=8$ for specificvalues of potential $\phi$ in the grand canonical ensemble. Furthermore, weobtain a similar behavior for the limit of $\beta \to \infty$, i.e charged-AdSblack holes in the third order of the Lovelock gravity. Thus, it is shown thatthe critical behaviors of these black holes are independent of the parameter$\beta$ in the grand canonical ensemble.

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