  Interpreting the cosmological constant as a thermodynamic pressure and its
conjugate quantity as a thermodynamic volume, we reconsider the investigation
of P-V critical behaviors of (1+n)-dimensional topological AdS black holes in
Lovelock-Born-Infeld gravity. In particular, we give an explicit expression of
the universal number \chi=\frac{P_c v_c}{T_c} in terms of the space dimension
$n$. Then, we examine the phase transitions at the critical points of such
topological black holes for 6 \leq n \leq 11 as required by the physical
condition of the thermodynamical quantities. More precisely, the Ehrenfest
equations have been checked revealing that the black hole system undergoes a
second phase transition at the critical points.
