  We investigate the phase space of topological black hole solutions of
${\mathfrak {su}}(N)$ 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 $\omega_{j}$, $j=1,\ldots , N-1$. For ${\mathfrak
{su}}(2)$ gauge group, the function $\omega_{1}$ 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 $\omega_{j}$ functions has one
or more zeros. For most of the solutions, all the $\omega_{j}$ functions have
no zeros, and at least some of these are linearly stable.
