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Phase transitions in charged topological black holes dressed with a scalarfield are studied. These black holes are solutions of the Einstein-Maxwelltheory with a negative cosmological constant and a conformally coupled realself-interacting scalar field. Comparing, in the grand canonical ensemble, thefree energies of the hairy and undressed black holes two different phasetransitions are found. The first of them is one of second-order type and itoccurs at a temperature defined by the value of the cosmological constant.Below this temperature an undressed black hole spontaneously acquires a scalarhair. The other phase transition is one of first-order type. The correspondingcritical temperature, which is bounded from above by the one of the previouscase, strongly depends on the coupling constant of the quartic self-interactionpotential, and this transition only appears when the coupling constant is lessthan a certain value. In this case, below the critical temperature theundressed black is thermodynamically favored. However, when the temperatureexceeds the critical value a hairy black hole is likely to be occur.

Phase transitions in charged topological black holes dressed with a scalar hair