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We employ a thermodynamic analysis to determine the phase structure ofEguchi-Hanson solitons, Schwarzschild-AdS/$\mathbb{Z}_k$ black holes andthermal AdS/$\mathbb{Z}_k$. The Euclidean actions are calculated by two equablemeans: the first uses the Eguchi-Hanson soliton as the thermal background whilethe second makes use of minimal boundary counterterms in the action necessaryto render individual actions finite. The Euclidean actions are then utilised todetermine the phase structure in arbitrary odd dimension; it is found thatthere is a Hawking-Page phase transition and also a phase transition betweenthe black hole and soliton. There is found to be no smooth phase transitiongoverned by an order parameter between AdS/$\mathbb{Z}_k$ and the soliton butnevertheless AdS/$\mathbb{Z}_k$ changes phase by tunneling to the lower energysoliton configuration.

Phase Transitions Between Solitons and Black Holes in Asymptotically AdS/$Z_k$ Spaces