We study the r31=LCO(3-2)/LCO(1-0) luminosity line ratio in a sample of nearby (z < 0.05) galaxies: 25 star-forming galaxies (SFGs) from the xCOLD GASS survey, 36 hard X-ray selected AGN host galaxies from BASS and 37 infrared luminous galaxies from SLUGS. We find a trend for r31 to increase with star-formation efficiency (SFE). We model r31 using the UCL-PDR code and find that the gas density is the main parameter responsible for variation of r31, while the interstellar radiation field and cosmic ray ionization rate play only a minor role. We interpret these results to indicate a relation between SFE and gas density. We do not find a difference in the r31 value of SFGs and AGN host galaxies, when the galaxies are matched in SSFR (<r31>= 0.52 +/- 0.04 for SFGs and <r31> = 0.53 +/- 0.06 for AGN hosts). According to the results of UCL-PDR models, the X-rays can contribute to the enhancement of the CO line ratio, but only for strong X-ray fluxes and for high gas density (nH > 10$^4$ cm-3). We find a mild tightening of the Kennicutt-Schmidt relation when we use the molecular gas mass surface density traced by CO(3-2) (Pearson correlation coefficient R=0.83), instead of the molecular gas mass surface density traced by CO(1-0) (R=0.78), but the increase in correlation is not statistically significant (p-value=0.06). This suggests that the CO(3-2) line can be reliably used to study the relation between SFR and molecular gas for normal SFGs at high redshift, and to compare it with studies of low-redshift galaxies, as is common practice.
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