  Clusters of galaxies are the most impressive gravitationally-bound systems in
the Universe and its abundance (the cluster mass function) is one important
statistics to probe the matter density parameter ($\Omega_m$) and the amplitude
of density fluctuations ($\sigma_8$). The cluster mass function is usually
described in terms of the Press-Schecther (PS) formalism where the primordial
density fluctuations are assumed to be a Gaussian random field. In previous
works we have proposed a non-Gaussian analytical extension of the PS approach
with basis on the $q$-power law distribution (PL) of the non-extensive kinetic
theory. In this paper, by applying the PL distribution to fit the observational
mass function data from X-ray highest flux-limited sample (HIFLUGCS) we find a
strong degeneracy among the cosmic parameters, $\sigma_8$, $\Omega_m$, and the
$q$ parameter from the PL distribution. A joint analysis involving recent
observations from baryon acoustic oscillation (BAO) peak and Cosmic Microwave
Background (CMB) shift parameter is carried out in order to break these
degeneracy and better constrain the physically relevant parameters. The present
results suggest that the next generation of cluster surveys will be able to
probe the quantities of cosmological interest ($\sigma_8, \Omega_m$) and the
underlying cluster physics quantified by the $q$-parameter.
