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.
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