  Even though the abundance and evolution of clusters have been used to study
the cosmological parameters including the properties of dark energy owing to
their pure dependence on the geometry of the Universe and the power spectrum,
it is necessary to pay particular attention to the effects of dark energy on
the analysis. We obtain the explicit dark energy dependent {\it rms} linear
mass fluctuation $\sigma_8$ which is consistent with the CMB normalization with
less than $2$ % errors for general constant dark energy equation of state,
$\oQ$. Thus, we do not have any degeneracy between $\sigma_8$ and the matter
energy density contrast $\Omo$. When we use the correct value of the critical
density threshold $\delta_{c} = 1.58$ obtained recently \cite{09090826,
09100126} into the cluster number density $n$ calculation in the
Press-Schechter (PS) formalism, $n$ increases as compared to the one obtained
by using $\delta_{c} = 1.69$ by about $60$, $80$, and $110$ % at $z = 0$,
$0.5$, and $1$, respectively. Thus, PS formalism predicts the cluster number
consistent with both simulation and observed data at the high mass region. We
also introduce the improved coefficients of Sheth-Tormen (ST) formalism, which
is consistent with the recently suggested mass function \cite{10052239}. We
found that changing $\oQ$ by $\Delta \oQ = -0.1$ from $\oQ = -1.0$ causes the
changing of the comoving numbers of high mass clusters of $M = 10^{16} h^{-1}
M_{\odot}$ by about $20$ and $40$ % at $z = 0$ and $1$, respectively.
