  We study the stability of static, hyperbolic Gauss-Bonnet black holes in
(4+1)-dimensional Anti-de-Sitter (AdS) space-time against the formation of
scalar hair. Close to extremality the black holes possess a near-horizon
topology of AdS_2 x H^3 such that within a certain range of the scalar field
mass one would expect that they become unstable to the condensation of an
uncharged scalar field. We confirm this numerically and observe that there
exists a family of hairy black hole solutions labelled by the number of nodes
of the scalar field function. We construct explicit examples of solutions with
a scalar field that possesses zero nodes, one node and two nodes, respectively,
and show that the solutions with nodes persist in the limit of Einstein
gravity, i.e. for vanishing Gauss-Bonnet coupling. We observe that the interval
of the mass for which scalar field condensation appears decreases with
increasing Gauss-Bonnet coupling and/or with increasing node number.
