  We present a novel approach to constrain accelerating cosmologies with galaxy
cluster phase spaces. With the Fisher matrix formalism we forecast constraints
on the cosmological parameters that describe the cosmological expansion
history. We find that our probe has the potential of providing constraints
comparable to, or even stronger than, those from other cosmological probes.
More specifically, with 1000 (100) clusters uniformly distributed in the
redshift range $ 0 \leq z \leq 0.8$, after applying a conservative $80\%$ mass
scatter prior on each cluster and marginalizing over all other parameters, we
forecast $1\sigma$ constraints on the dark energy equation of state $w$ and
matter density parameter $\Omega_M$ of $\sigma_w = 0.138 (0.431)$ and
$\sigma_{\Omega_M} = 0.007 (0.025)$ in a flat universe. Assuming 40\% mass
scatter and adding a prior on the Hubble constant we can achieve a constraint
on the CPL parametrization of the dark energy equation of state parameters
$w_0$ and $w_a$ with 100 clusters in the same redshift range: $\sigma_{w_0} =
0.191 $ and $\sigma_{w_a} = 2.712$. Dropping the assumption of flatness and
assuming $w=-1$ we also attain competitive constraints on the matter and dark
energy density parameters: $\sigma_{\Omega_M} = 0.101$ and
$\sigma_{\Omega_{\Lambda}} = 0.197$ for 100 clusters uniformly distributed in
the range $ 0 \leq z \leq 0.8$ after applying a prior on the Hubble constant.
We also discuss various observational strategies for tightening constraints in
both the near and far future.
