We study the quark deconfinement phase transition in cold (T = 0) and hot beta-stable hadronic matter. Assuming a first-order phase transition, we calculate and compare the nucleation rate and the nucleation time due to thermal and quantum nucleation mechanisms. We show that above a threshold value of the central pressure a pure hadronic star (HS) is metastable to the conversion to a quark star (QS) (i.e. hybrid star or strange star). We introduce the concept of critical mass M(cr) for cold HSs and proto-hadronic stars, and the concept of limiting conversion temperature for proto-hadronic stars. We show that proto-hadronic stars with a mass M < M(cr) could survive the early stages of their evolution without decaying to QSs. Finally, we discuss the possible evolutionary paths of proto-hadronic stars.