We investigate the phase space of topological black hole solutions of su(N) Einstein–Yang–Mills theory in anti-de Sitter space with a purely magnetic gauge potential. The gauge field is described by N−1 magnetic gauge field functions ωj, j=1,…,N−1. For su(2) gauge group, the function ω1 has no zeros. This is no longer the case when we consider a larger gauge group. The phase space of topological black holes is considerably simpler than for the corresponding spherically symmetric black holes, but for N>2 and a flat event horizon, there exist solutions where at least one of the ωj functions has one or more zeros. For most of the solutions, all the ωj functions have no zeros, and at least some of these are linearly stable.