  We construct a SCUBA-2 450-$\mu$m map in the COSMOS field that covers an area
of 300 arcmin$^{2}$ and reaches a 1$\sigma$ noise level of 0.65 mJy in the
deepest region. We extract 256 sources detected at 450 $\mu$m with
signal-to-noise ratio $>$ 4.0 and analyze the physical properties of their
multi-wavelength counterparts. We find that most of the sources are at
$z\lesssim3$, with a median of $z = 1.79^{+0.03}_{-0.15}$. About
$35^{+32}_{-25}$% of our sources are classified as starburst galaxies based on
their total star-formation rates (SFRs) and stellar masses ($M_{\ast}$). By
fitting the far-infrared spectral energy distributions, we find that our
450-$\mu$m-selected sample has a wide range of dust temperatures (20 K $
\lesssim T_{\rm d} \lesssim$ 60 K), with a median of ${T}_{\rm d} =
38.3^{+0.4}_{-0.9}$ K. We do not find a redshift evolution in dust temperature
for sources with $L_{\rm IR}$ > $10^{12}$ $\rm L_\odot$ at $z<3$. However, we
find a moderate correlation where dust temperature increases with the deviation
from the SFR-$M_{\ast}$ relation. The increase in dust temperature also
correlates with optical morphology, which is consistent with merger-triggered
starbursts in sub-millimeter galaxies. Our galaxies do not show the tight
IRX-$\beta_{\rm UV}$ correlation that has been observed in the local Universe.
We construct the infrared luminosity functions of our 450-$\mu$m sources and
measure their comoving SFR densities. The contribution of the $L_{\rm IR}$ >
$10^{12}$ $\rm L_\odot$ population to the SFR density rises dramatically from
$z$ = 0 to 2 ($\propto$ ($1+z$)$^{3.9\pm1.1}$) and dominates the total SFR
density at $z \gtrsim 2$.
