  We study the equation of state (EOS) of kaon-condensed matter including the
effects of temperature and trapped neutrinos. It is found that the order of the
phase transition to a kaon-condensed phase, and whether or not Gibbs' rules for
phase equilibrium can be satisfied in the case of a first order transition,
depend sensitively on the choice of the kaon-nucleon interaction. The main
effect of finite temperature, for any value of the lepton fraction, is to mute
the effects of a first order transition, so that the thermodynamics becomes
similar to that of a second order transition. Above a critical temperature,
found to be at least 30--60 MeV depending upon the interaction, the first order
transition disappears. The phase boundaries in baryon density versus lepton
number and baryon density versus temperature planes are delineated. We find
that the thermal effects on the maximum gravitational mass of neutron stars are
as important as the effects of trapped neutrinos, in contrast to previously
studied cases in which the matter contained only nucleons or in which hyperons
and/or quark matter were considered. Kaon-condensed EOSs permit the existence
of metastable neutron stars, because the maximum mass of an initially hot,
lepton-rich protoneutron star is greater than that of a cold, deleptonized
neutron star. The large thermal effects imply that a metastable protoneutron
star's collapse to a black hole could occur much later than in previously
studied cases that allow metastable configurations.
