In order to precisely determine temperature and density of molecular gas in the Large Magellanic Cloud, we made observations of optically thin $^{13}$CO($J=3-2$) transition by using the ASTE 10m telescope toward 9 peaks where $^{12}$CO($J=3-2$) clumps were previously detected with the same telescope. The molecular clumps include those in giant molecular cloud (GMC) Types I (with no signs of massive star formation), II (with HII regions only), and III (with HII regions and young star clusters). We detected $^{13}$CO($J=3-2$) emission toward all the peaks and found that their intensities are 3 -- 12 times lower than those of $^{12}$CO($J=3-2$). We determined the intensity ratios of $^{12}$CO($J=3-2$) to $^{13}$CO($J=3-2$), $R^{12/13}_{3-2}$, and $^{13}$CO($J=3-2$) to $^{13}$CO($J=1-0$), $R^{13}_{3-2/1-0}$, at 45$arcsec$ resolution. These ratios were used for radiative transfer calculations in order to estimate temperature and density of the clumps. The parameters of these clumps range kinetic temperature $Tmathrm{_{kin}}$ = 15 -- 200 K, and molecular hydrogen gas density $n(mathrm{H_2})$ = 8$times 10^2$ -- 7$times 10^3$ cm$^{-3}$. We confirmed that the higher density clumps show higher kinetic temperature and that the lower density clumps lower kinetic temperature at a better accuracy than in the previous work. The kinetic temperature and density increase generally from a Type I GMC to a Type III GMC. We interpret that this difference reflects an evolutionary trend of star formation in molecular clumps. The $R^{13}_{3-2/1-0}$ and kinetic temperature of the clumps are well correlated with H$alpha$ flux, suggesting that the heating of molecular gas $n(mathrm{H_2})$ = $10^3$ -- $10^4$ cm$^{-3}$ can be explained by stellar FUV photons.
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