  The influence of considering a generalized dark matter (GDM) model, which
allows for a non-pressure-less dark matter and a non-vanishing sound speed in
the non-linear spherical collapse model is discussed for the Einstein-de
Sitter-like (EdSGDM) and $\Lambda$GDM models. By assuming that the vacuum
component responsible for the accelerated expansion of the Universe is not
clustering and therefore behaving similarly to the cosmological constant
$\Lambda$, we show how the change in the GDM characteristic parameters affects
the linear density threshold for collapse of the non-relativistic component
($\delta_{\rm c}$) and its virial overdensity ($\Delta_{\rm V}$). We found that
the generalized dark matter equation of state parameter $w_{\rm gdm}$ is
responsible for lower values of the linear overdensity parameter as compared to
the standard spherical collapse model and that this effect is much stronger
than the one induced by a change in the generalized dark matter sound speed
$c^2_{\rm s, gdm}$. We also found that the virial overdensity is only slightly
affected and mostly sensitive to the generalized dark matter equation of state
parameter $w_{\rm gdm}$. These effects could be relatively enhanced for lower
values of the matter density. Finally, we found that the effects of the
additional physics on $\delta_{\rm c}$ and $\Delta_{\rm V}$, when translated to
non-linear observables such as the halo mass function, induce an overall
deviation of about 40\% with respect to the standard $\Lambda$CDM model at late
times for high mass objects. However, within the current linear constraints for
$c^2_{\rm s, gdm}$ and $w_{\rm gdm}$, we found that these changes are the
consequence of properly taking into account the correct linear matter power
spectrum for the GDM model while the effects coming from modifications in the
spherical collapse model remain negligible.
