  It has been found in previous studies that, for the Supernova Legacy Survey
three-year (SNLS3) data, there is strong evidence for the redshift-evolution of
color-luminosity parameter $\beta$. In this paper, using three simplest dark
energy models ($\Lambda$CDM, $w$CDM, and CPL), we further explore the evolution
of $\beta$ and its effects on parameter estimation. In addition to the SNLS3
data, we also take into account the Planck distance priors data, as well as the
latest galaxy clustering (GC) data extracted from SDSS DR7 and BOSS. We find
that, for all the models, adding a parameter of $\beta$ can reduce
$\chi^2_{min}$ by $\sim$ 36, indicating that $\beta_1 = 0$ is ruled out at
6$\sigma$ confidence levels. In other words, $\beta$ deviates from a constant
at 6$\sigma$ confidence levels. This conclusion is insensitive to the dark
energy models considered, showing the importance of considering the evolution
of $\beta$ in the cosmology-fits. Furthermore, it is found that varying $\beta$
can significantly change the fitting results of various cosmological
parameters: using the SNLS3 data alone, varying $\beta$ yields a larger
$\Omega_m$ for the $\Lambda$CDM model; using the SNLS3+CMB+GC data, varying
$\beta$ yields a larger $\Omega_m$ and a smaller $h$ for all the models.
Moreover, we find that these results are much closer to those given by the
CMB+GC data, compared to the cases of treating $\beta$ as a constant. This
indicates that considering the evolution of $\beta$ is very helpful for
reducing the tension between supernova and other cosmological observations.
