Thermodynamics of black holes with an infinite effective area of a horizon

In some kinds of classical dilaton theory there exist black holes with (i) infinite horizon area A or infinite F (the coefficient at curvature in the Lagrangian) and (ii) zero Hawking temperature TH. For a generic static black hole, without an assumption about spherical symmetry, we show that infinite A is compatible with a regularity of geometry in the case TH = 0 only. We also point out that infinite TH is incompatible with the regularity of a horizon of a generic static black hole, both for finite or infinite A. A direct application of the standard Euclidean approach in the case of an infinite 'effective' area of the horizon Aeff = AF leads to inconsistencies in the variational principle and gives an indefinite expression for black-hole entropy S, formally proportional to THAeff. We show that treating a horizon as an additional boundary (that is, adding to the action some terms calculated on the horizon) may restore self-consistency of the variational procedure, if F does not grow too rapidly near the horizon. We apply this approach to Brans-Dicke black holes and obtain the same answer S = 0 as for 'usual' (e.g., Reissner-Nordstroem) extreme classical black holes. We also consider the exact solution for a conformal coupling, when A is finite but F diverges and find that in the latter case both the standard and modified approaches give rise to an infinite action. Thus, this solution represents a rare exception of a black hole without non-trivial thermal properties