Abbott–Deser–Tekin Charge of Dilaton Black Holes with Squashed Horizons

We consider the conserved charge of static black holes with squashed horizons in the Einstein–Maxwell–dilaton theory via both the Abbott–Deser–Tekin (ADT) method and its off-shell generalization. We first make use of the original ADT method to compute the mass of the dilaton squashed black holes in terms of three different reference spacetimes, which are the asymptotic geometry, the flat background and the spacetime of the Kaluza–Klein monopole with boundary matched to the original metric, respectively. Each mass satisfies the first law of black hole thermodynamics, although the mass computed on the basis of the boundary matching the Kaluza–Klein monopole is different from that of the other two reference spacetimes. Then the mass of the black holes is evaluated through the off-shell generalized ADT method.