Thermodynamics of squashed Kaluza-Klein black holes and black strings: a comparison of reference backgrounds

We investigate thermodynamics constructed on different background reference spacetimes for squashed Kaluza-Klein (SqKK) black hole and electrically charged black string in the five-dimensional Einstein-Maxwell system. Two spacetimes are possible to be reference spacetimes giving finite gravitational classical actions: one is four-dimensional Minkowski times a circle and the other is the KK monopole. The boundary of the SqKK black hole cannot be matched perfectly to that of the former reference spacetime because of the difference in topology. However, the resultant classical action coincides with that calculated by the counterterm subtraction scheme. The boundary of the KK monopole has the same topology as that of the SqKK black hole and can be matched to the boundary of the black hole perfectly. The resultant action takes a different value from the result given by using the former reference spacetime. After a brief review of thermodynamic quantities of the black hole solutions, we calculate thermodynamic potentials relevant for several thermodynamic environments. The most stable state is different for each environment: for example, the KK monopole is the most stable state in an isothermal environment with fixed gravitational tension. On the other hand, when the size of the extra dimension is fixed, the Minkowski times a circle is the most stable. It is shown that these two spacetimes can be reference spacetimes of the five-dimensional black string.