  The abundance of clusters of galaxies is known to be a potential source of
cosmological constraints through their mass function. In the present work, we
examine the information that can be obtained from the temperature distribution
function of X-ray clusters. For this purpose, the mass-temperature ($M$-$T$)
relation and its statistical properties are critical ingredients. Using a
combination of cosmic microwave background (CMB) data from Planck and our
estimations of X-ray cluster abundances, we use Markov chain Monte Carlo (MCMC)
techniques to estimate the $\Lambda$CDM cosmological parameters and the mass to
X-ray temperature scaling relation simultaneously. We determine the integrated
X-ray temperature function of local clusters using flux-limited surveys. A
local comprehensive sample was build from the BAX X-ray cluster database,
allowing us to estimate the local temperature distribution function above
$\sim$1 keV. We model the expected temperature function from the mass function
and the $M$-$T$ scaling relation. We then estimate the cosmological parameters
and the parameters of the $M$-$T$ relation (calibration and slope)
simultaneously. The measured temperature function of local clusters in the
range $\sim\!\!1$-$10$ keV is well reproduced once the calibration of the
$M$-$T$ relation is treated as a free parameter, and therefore is
self-consistent with respect to the $\Lambda$CDM cosmology. The best-fit values
of the standard cosmological parameters as well as their uncertainties are
unchanged by the addition of clusters data. The calibration of the mass
temperature relation, as well as its slope, are determined with $\sim10\%$
statistical uncertainties. This calibration leads to masses that are
$\sim\!\!75\%$ larger than X-ray masses used in Planck.
