  We study thermodynamics and critical behaviors of higher-dimensional Lovelock
black holes with non-maximally symmetric horizons in the canonical ensemble of
extended phase space. The effects from non-constancy of the horizon of the
black hole via appearing two chargelike parameters in thermodynamic quantities
of third-order Lovelock black holes are investigated. We find that Ricci flat
black holes with nonconstant curvature horizon show critical behavior. This is
an interesting feature that is not seen for any kind of black hole in Einstein
or Lovelock gravity in the literature. We examine how various interesting
thermodynamic phenomena such as standard first-order small-large black hole
phase transition, a reentrant phase transition, or zeroth order phase
transition happens for Ricci flat, spherical, or hyperbolic black holes with
nonconstant curvature horizon depending on the values of Lovelock coefficient
and chargelike parameters. While for a spherical black hole of third order
Lovelock gravity with constant curvature horizon phase transition is observed
only for $7\leq d \leq11$, for our solution criticality and phase transition
exist in every dimension. With a proper choice of the free parameters, a
large-small-large black hole phase transition occurs. This process is
accompanied by a finite jump of the Gibbs free energy referred to as a
zeroth-order phase transition. For the case $\kappa=-1$ a novel behavior is
found for which three critical points could exist.
