  We study the quark deconfinement phase transition in hot $\beta$-stable
hadronic matter. Assuming a first order phase transition, we calculate the
enthalpy per baryon of the hadron-quark phase transition. We calculate and
compare the nucleation rate and the nucleation time due to thermal and quantum
nucleation mechanisms. We compute the crossover temperature above which thermal
nucleation dominates the finite temperature quantum nucleation mechanism. We
next discuss the consequences for the physics of proto-neutron stars. We
introduce the concept of limiting conversion temperature and critical mass
$M_{cr}$ for proto-hadronic stars, and we show that proto-hadronic stars with a
mass $M < M_{cr}$ could survive the early stages of their evolution without
decaying to a quark star.
