  We explore the classical stability of topological black holes in
d-dimensional anti-de Sitter spacetime, where the horizon is an Einstein
manifold of negative curvature. According to the gauge invariant formalism of
Ishibashi and Kodama, gravitational perturbations are classified as being of
scalar, vector, or tensor type, depending on their transformation properties
with respect to the horizon manifold. For the massless black hole, we show that
the perturbation equations for all modes can be reduced to a simple scalar
field equation. This equation is exactly solvable in terms of hypergeometric
functions, thus allowing an exact analytic determination of potential
gravitational instabilities. We establish a necessary and sufficient condition
for stability, in terms of the eigenvalues $\lambda$ of the Lichnerowicz
operator on the horizon manifold, namely $\lambda \geq -4(d-2)$. For the case
of negative mass black holes, we show that a sufficient condition for stability
is given by $\lambda \geq -2(d-3)$.
