  We investigate the molecular gas properties of galaxies across the main
sequence of star-forming (SF) galaxies in the local Universe using
$^{12}$CO($J=1-0$) (hereafter $^{12}$CO) and $^{13}$CO($J=1-0$) ($^{13}$CO)
mapping data of 147 nearby galaxies obtained in the COMING project, a legacy
project of the Nobeyama Radio Observatory. In order to improve signal-to-noise
ratios of both lines, we stack all the pixels where $^{12}$CO emission is
detected after aligning the line center expected from the first-moment map of
$^{12}$CO. As a result, $^{13}$CO emission is successfully detected in 80
galaxies with a signal-to-noise ratio larger than three. The error-weighted
mean of integrated-intensity ratio of $^{12}$CO to $^{13}$CO lines ($R_{1213}$)
of the 80 galaxies is $10.9$ with a standard deviation of $7.0$. We find that
(1) $R_{1213}$ positively correlates to specific star-formation rate (sSFR)
with a correlation coefficient of $0.46$, and (2) both flux ratio of IRAS
60~$\mu$m to 100~$\mu$m ($f_{60}/f_{100}$) and inclination-corrected linewidth
of $^{12}$CO stacked spectra ($\sigma_{{\rm ^{12}CO},i}$) also correlate with
sSFR for galaxies with the $R_{1213}$ measurement. Our results support the
scenario where $R_{1213}$ variation is mainly caused by the changes in
molecular-gas properties such as temperature and turbulence. The consequent
variation of CO-to-H$_2$ conversion factor across the SF main sequence is not
large enough to completely extinguish the known correlations between sSFR and
$M_{\rm mol}/M_{\rm star}$ ($\mu_{\rm mol}$) or star-formation efficiency (SFE)
reported in previous studies, while this variation would strengthen (weaken)
the sSFR-SFE (sSFR-$\mu_{\rm mol}$) correlation.
