  We present a new class of black hole solutions in third-order Lovelock
gravity whose horizons are Einstein space with two supplementary conditions on
their Weyl tensors. These solutions are obtained with the advantage of higher
curvature terms appearing in Lovelock gravity. We find that while the solution
of third-order Lovelock gravity with constant-curvature horizon in the absence
of a mass parameter is the anti de Sitter (AdS) metric, this kind of solution
with nonconstant- curvature horizon is only asymptotically AdS and may have
horizon. We also find that one may have an extreme black hole with non-constant
curvature horizon whose Ricci scalar is zero or a positive constant, while
there is no such black hole with constant-curvature horizon. Furthermore, the
thermodynamics of the black holes in the two cases of constant- and
nonconstant-curvature horizons are different drastically. Specially, we
consider the thermodynamics of black holes with vanishing Ricci scalar and find
that in contrast to the case of black holes of Lovelock gravity with
constant-curvature horizon, the area law of entropy is not satisfied. Finally,
we investigate the stability of these black holes both locally and globally and
find that while the black holes with constant curvature horizons are stable
both locally and globally, those with nonconstant-curvature horizons have
unstable phases.
