  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
Lagrangian) and (ii) zero Hawking temperature $T_{H}$. 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 $T_{H}=0$
only. We also point out that infinite $T_{H}$ is incompatible with the
regularity of a horizon of a generic static black hole, both for finite or
infinite $A$. Direct application of the standard Euclidean approach in the case
of an infinite ''effective'' area of the horizon $A_{eff}=AF$ leads to
inconsistencies in the variational principle and gives for a black hole entropy
$S$ an indefinite expression, formally proportional to $T_{H}A_{eff}$. 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 $F$ near the horizon grows not too rapidly. We
apply this approach to Brans-Dicke black holes and obtain the same answer S=0
as for ''usual'' (for example, Reissner-Nordstr\"{o}m) 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 approach give rise to an infinite action. Thus, this solution
represents a rare exception of a black hole without nontrivial thermal
properties.
