  Treating the cosmological constant as thermodynamic pressure and its
conjugate as thermodynamic volume, we investigate the critical behavior of the
third order Lovelock black holes in diverse dimensions. For black hole horizons
with different normalized sectional curvature $k=0,\pm1$, the corresponding
critical behaviors differ drastically. For $k=0$, there is no critical point in
the extended thermodynamic phase space. For $k=-1$, there is a single critical
point in any dimension $d\geq 7$, and for $k=+1$, there is a single critical
point in $7$ dimension and two critical points in $8,9,10,11$ dimensions. We
studied the corresponding phase structures in all possible cases.
