P-V criticality of topological black holes in Lovelock-Born-Infeld gravity

To understand the effect of third order Lovelock gravity, - criticality of topological AdS black holes in Lovelock-Born-Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and in some more detail than the previous literature. A detailed analysis of the limit case is performed for the seven-dimensional black holes. It is shown that, for the spherical topology, - criticality exists for both the uncharged and the charged cases. Our results demonstrate again that the charge is not the indispensable condition of - criticality. It may be attributed to the effect of higher derivative terms of the curvature because similar phenomenon was also found for Gauss-Bonnet black holes. For , there would be no - criticality. Interesting findings occur in the case , in which positive solutions of critical points are found for both the uncharged and the charged cases. However, the - diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of , the entropy is always positive for any specific volume . Since no - criticality exists for in Einstein gravity and Gauss-Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss-Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and swallow tail behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research.