In a first-order cosmological phase transition, the tunneling from the false to the true vacuum (see left picture below) releases false vacuum energy corresponding to the potential difference $\Delta V$ between the two different vacua.
In a continuous (second-order / cross-over) cosmological phase transition, there is also a potential difference $\Delta V$ between the initial and final vacua (see right picture below). Does this imply that a continuous phase transition also releases false vacuum energy?
If so, is the equation-of-state parameter of this vacuum energy $\omega=-1$, just as in the first-order case? Is the only difference to the first-order case that the vacuum energy now gets slowly released?
If the answers to the above questions are "yes", does this imply that the very early Universe (before the Higgs and QCD phase transitions) contained a constant amount of vacuum energy, which got successively released in the different phase transitions? Would there be any potentially observable consequences, or would this vacuum energy be negligible compared to the energy stored in radiation?
Finally, what does the false vacuum energy convert into? Most textbooks say "radiation", but if we consider for example the Higgs phase transition, the vacuum expectation value of the Higgs field gives rise to fermion masses, so shouldn't the vacuum energy partially convert into fermion masses as well?
Time evolution of effective potential $V(\phi)$ in first-order (left) and continuous (right) phase transitions.