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 ΛCDM model. The comparison with ΛCDM is performed by a two-step process. First, we determine the free parameters of all models through a joint analysis involving the latest cosmological data, using supernovae type Ia, the cosmic microwave background shift parameter, and baryon acoustic oscillations. Apart from a braneworld inspired cosmology, it is found that the derived Hubble relation of the remaining models reproduces the ΛCDM results approximately with the same degree of statistical confidence. Second, in order to attempt to distinguish the different dark energy models from the expectations of ΛCDM, 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 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 ΛCDM cosmology, while the predictions of 4 models are statistically equivalent to those of the ΛCDM model, as far as the expected cluster mass function and redshift distribution are concerned. The present analysis suggests 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.