  By a simple modification of Hawking's well-known topology theorems for black
hole horizons, we find lower bounds for the areas of smooth apparent horizons
and smooth cross-sections of stationary black hole event horizons of genus
$g>1$ in four dimensions. For a negatively curved Einstein space, the bound is
${{4\pi (g-1)}\over {-\ell}}$ where $\ell$ is the cosmological constant of the
spacetime. This is complementary to the known upper bound on the area of $g=0$
black holes in de Sitter spacetime. It also emerges that $g>1$ quite generally
requires a mean negative energy density on the horizon. The bound is sharp; we
show that it is saturated by certain extreme, asymptotically locally anti-de
Sitter spacetimes. Our results generalize a recent result of Gibbons.
