  Galaxy cluster number count has been proven to be a powerful cosmological
probe. However, cosmological constraints established with galaxy cluster number
count are highly dependent on the calibration of the mass-observable relations.
Thanks to its nearly mass independence the specific mass accretion rate of
galaxy clusters is nearly insensitive to the calibration of mass-observable
relations. The study of galaxy cluster number count evolution allows to probe
the galaxy cluster mass accretion history in the context of an homogenous
Universe. In this paper, we use relative abundance matching to infer the galaxy
cluster mass accretion rate (MAR) for $z \in [0.0,0.6[$. Then, we use the MAR
to set cosmological constraints. We found that this cosmological probe is
sensitive to $\sigma_8 \Omega_{\rm m}^{-0.3} H_0^{-0.2}$ whereas the galaxy
cluster count is sensitive to $\sigma_8 \Omega_{\rm m}^{0.3}$. We used the
second $Planck$ catalog of Sunyaev-Zel'dovich sources and we derive $\sigma_8
\Omega_{\rm m}^{-0.3} H_0^{-0.2} = 0.75 \pm 0.06$. This results is consistent
with cosmological constraints derived from galaxy clusters number counts,
angular power spectrum, and cosmic microwave background analyses. Therefore,
the MAR is a key cosmological probe that can break the $\sigma_8$-$\Omega_{\rm
m}$ degeneracy and that is not sensitive to the calibration of the
mass-observable relations and does not requires a parametric form for the
galaxy cluster mass-function.
