  We investigate the linear stability of topological black hole solutions to
four-dimensional SU(N) Einstein-Yang-Mills theory with a negative cosmological
constant. We here extend recent results in the field which prove the existence
of hairy black hole solutions to such equations, and the stability of their
spherically symmetric analogues. We find the analysis carries over very
similarly, with some important differences in the final stages. Nevertheless,
we establish the existence of non-trivial solutions stable under linear
perturbations, in a sufficiently small neighbourhood of some existing trivial
solutions; in fact, stable topological solutions turn out to be likely more
abundant in the parameter space than their spherically symmetric analogues.
