We investigate the physical properties of the protoneutron stars in the framework of a relativistic mean-field model and we study the finite-temperature equation of state in β-stable matter at fixed entropy per baryon, in the presence of hyperons, Δ-isobar resonances and trapped neutrinos. In this context, we study the possible presence of thermodynamic instabilities and a phase transition from nucleonic matter to resonance-dominated Δ-matter can take place. Such a phase transition is characterized by both mechanical instability (fluctuations on the baryon density) and by chemical-diffusive instability (fluctuations on the isospin concentration) in asymmetric nuclear matter. We show that such statistical effects could play a crucial role in the structure and in the evolution of the protoneutron stars.