  The mass function of cluster-size halos and their redshift distribution are
computed for 12 distinct accelerating cosmological scenarios and confronted to
the predictions of the conventional flat $\Lambda$CDM model. The comparison
with $\Lambda$CDM is performed by a two-step process. Firstly, we determine the
free parameters of all models through a joint analysis involving the latest
cosmological data, using SNe type Ia, the CMB shift parameter and BAO. Apart
from a brane world inspired cosmology, it is found that the derived Hubble
relation of theremaining models reproduce the $\Lambda$CDM results
approximately with the same degree of statistical confidence. Secondly, in
order to attempt distinguish the different dark energy models from the
expectations of $\Lambda$CDM, we analyze the predicted cluster-size halo
redshift distribution on the basis of two future cluster surveys: (i) an X-ray
survey based on the {\tt eROSITA} satellite, and (ii) a Sunayev-Zeldovich
survey based on the South Pole Telescope. As a result, we find that the
predictions of 8 out of 12 dark energy models can be clearly distinguished from
the $\Lambda$CDM cosmology, while the predictions of 4 models are statistically
equivalent to those of the $\Lambda$CDM model, as far as the expected cluster
mass function and redshift distribution are concerned. The present analysis
suggest that such a technique appears to be very competitive to independent
tests probing the late time evolution of the Universe and the associated dark
energy effects.
