  [Abridged] We use data on massive galaxy clusters ($M_{\rm cluster} > 8
\times 10^{14} h^{-1} M_\odot$ within a comoving radius of $R_{\rm cluster} =
1.5 h^{-1}\Mpc$) in the redshift range $0.05 \lesssim z \lesssim 0.83$ to place
constraints, simultaneously, on the nonrelativistic matter density parameter
$\Omega_m$, on the amplitude of mass fluctuations $\sigma_8$, on the index $n$
of the power-law spectrum of the density perturbations, and on the Hubble
constant $H_0$, as well as on the equation-of-state parameters $(w_0,w_a)$ of a
smooth dark energy component.
  For the first time, we properly take into account the dependence on redshift
and cosmology of the quantities related to cluster physics: the critical
density contrast, the growth factor, the mass conversion factor, the virial
overdensity, the virial radius and, most importantly, the cluster number count
derived from the observational temperature data.
  We show that, contrary to previous analyses, cluster data alone prefer low
values of the amplitude of mass fluctuations, $\sigma_8 \leq 0.69 (1\sigma
C.L.)$, and large amounts of nonrelativistic matter, $\Omega_m \geq 0.38
(1\sigma C.L.)$, in slight tension with the $\Lambda$CDM concordance
cosmological model, though the results are compatible with $\Lambda$CDM at
$2\sigma$. In addition, we derive a $\sigma_8$ normalization relation,
$\sigma_8 \Omega_m^{1/3} = 0.49 \pm 0.06 (2\sigma C.L.)$.
