We study the equation of state (EOS) of kaon-condensed matter including theeffects of temperature and trapped neutrinos. It is found that the order of thephase transition to a kaon-condensed phase, and whether or not Gibbs' rules forphase equilibrium can be satisfied in the case of a first order transition,depend sensitively on the choice of the kaon-nucleon interaction. The maineffect of finite temperature, for any value of the lepton fraction, is to mutethe effects of a first order transition, so that the thermodynamics becomessimilar 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 ordertransition disappears. The phase boundaries in baryon density versus leptonnumber and baryon density versus temperature planes are delineated. We findthat the thermal effects on the maximum gravitational mass of neutron stars areas important as the effects of trapped neutrinos, in contrast to previouslystudied cases in which the matter contained only nucleons or in which hyperonsand/or quark matter were considered. Kaon-condensed EOSs permit the existenceof metastable neutron stars, because the maximum mass of an initially hot,lepton-rich protoneutron star is greater than that of a cold, deleptonizedneutron star. The large thermal effects imply that a metastable protoneutronstar's collapse to a black hole could occur much later than in previouslystudied cases that allow metastable configurations.
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