  The generalization of Birkhoff's theorem to higher dimensions in Lovelock
gravity allows for black hole solutions with horizon geometries of non-constant
curvature. We investigate thermodynamic aspects of these `exotic' black hole
solutions, with a particular emphasis on their phase transitions. We consider
an extended phase space where the cosmological constant acts as a thermodynamic
pressure, and examine both uncharged and $U(1)$ charged solutions. In $d=7$,
black hole solutions are restricted to having constant-curvature horizon base
manifolds. Uncharged $d=7$ black holes possess novel triple point phenomena
analogous to those recently uncovered in exotic $d=6$ black holes in
Gauss-Bonnet gravity, while their charged counterparts generically undergo
small-large black hole phase transitions. In $d=8$, we find that both charged
and uncharged black holes exhibit triple point behaviour and small-large black
hole transitions. We also show that a wide range of `exotic' horizon geometries
can be ruled out due to the appearance of naked singularities.
