Abstract
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 ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{H}^{3}$ 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 labeled 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.
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