  The study of the primordial black hole (PBH) gravitational collapse process
requires the determination of a critical energy density perturbation threshold
$\delta_\mathrm{c}$, which depends on the equation of state of the universe at
the time of PBH formation. Up to now, the majority of analytical and numerical
techniques calculate $\delta_\mathrm{c}$ by assuming a constant
equation-of-state (EoS) parameter $w$ at the time of PBH formation. In this
work, after generalizing the constant $w$ prescription of [1] for the
computation of $\delta_\mathrm{c}$ and commenting its limitations we give a
first estimate for the PBH threshold in the case of a time-dependent $w$
background. In particular, we apply our formalism in the case of the QCD phase
transition, where the EoS parameter varies significantly with time and one
expects an enhanced PBH production due to the abrupt softening of $w$. At the
end, we compare our results with analytic and numerical approaches for the
determination of $\delta_\mathrm{c}$ assuming a constant EoS parameter.
