We employ a thermodynamic analysis to determine the phase structure of Eguchi–Hanson solitons, Schwarzschild–AdS/Zk black holes and thermal AdS/Zk. The Euclidean actions are calculated by two equable means: the first uses the Eguchi–Hanson soliton as the thermal background while the second makes use of minimal boundary counterterms in the action necessary to render individual actions finite. The Euclidean actions are then utilised to determine the phase structure in arbitrary odd dimension; it is found that there is a Hawking–Page phase transition and also a phase transition between the black hole and soliton. There is found to be no smooth phase transition governed by an ozder parameter between AdS/Zk and the soliton but nevertheless AdS/Zk changes phase by tunneling to the lower energy soliton configuration.
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