  We study charged, static, topological black holes in pure Gauss-Bonnet
gravity in asymptotically AdS space. As in general relativity, the theory
possesses a unique nondegenerate AdS vacuum. It also admits charged black hole
solutions which asymptotically behave as the Reissner-Nordstr\"{o}m AdS black
hole. We discuss black hole thermodynamics of these black holes. Then we study
phase transitions in a dual quantum field theory in four dimensions, with the
St\"{u}ckelberg scalar field as an order parameter. We find in the probe limit
that the black hole can develop hair below some critical temperature, which
suggests a phase transition. Depending on the scalar coupling constants, the
phase transition can be first or second order. Analysis of the free energy
reveals that, comparing the two solutions, the hairy state is energetically
favorable, thus a phase transition will occur in a dual field theory.
