  The abundance of galaxy clusters is in principle a powerful tool to constrain
cosmological parameters, especially $\Omega_\mathrm{m}$ and $\sigma_8$, due to
the exponential dependence in the high-mass regime. While the best observables
are the X-ray temperature and luminosity, the abundance of galaxy clusters,
however, is conventionally predicted as a function of mass. Hence, the
intrinsic scatter and the uncertainties in the scaling relations between mass
and either temperature or luminosity lower the reliability of galaxy clusters
to constrain cosmological parameters. In this article, we further refine the
X-ray temperature function for galaxy clusters by Angrick et al., which is
based on the statistics of perturbations in the cosmic gravitational potential
and proposed to replace the classical mass-based temperature function, by
including a refined analytic merger model and compare the theoretical
prediction to results from a cosmological hydrodynamical simulation. Although
we find already a good agreement if we compare with a cluster temperature
function based on the mass-weighted temperature, including a redshift-dependent
scaling between mass-based and spectroscopic temperature yields even better
agreement between theoretical model and numerical results. As a proof of
concept, incorporating this additional scaling in our model, we constrain the
cosmological parameters $\Omega_\mathrm{m}$ and $\sigma_8$ from an X-ray sample
of galaxy clusters and tentatively find agreement with the recent cosmic
microwave background based results from the Planck mission at 1$\sigma$-level.
