  Optically-similar early-type galaxies are observed to have a large and poorly
understood range in the amount of hot, X-ray-emitting gas they contain.To
investigate the origin of this diversity, we studied the hot gas properties of
all 42 early-type galaxies in the multiwavelength ATLAS$^{\rm 3D}$ survey that
have sufficiently deep {\sl Chandra} X-ray observations. We related their hot
gas properties to a number of internal and external physical quantities. To
characterize the amount of hot gas relative to the stellar light, we use the
ratio of the gaseous X-ray luminosity to the stellar $K$-band luminosity,
$L_{X_{\rm gas}}/L_K$; we also use the deviations of $L_{X_{\rm gas}}$ from the
best-fit $L_{X_{\rm gas}}$--$L_K$ relation (denoted $\Delta L_{X_{\rm gas}}$).
We quantitatively confirm previous suggestions that various effects conspire to
produce the large scatter in the observed $L_X/L_K$ relation. In particular, we
find that the deviations $\Delta L_{X_{\rm gas}}$ are most strongly positively
correlated with the (low rates of) star formation and the hot gas temperatures
in the sample galaxies. This suggests that mild stellar feedback may energize
the gas without pushing it out of the host galaxies. We also find that galaxies
in high galaxy density environments tend to be massive slow-rotators, while
galaxies in low galaxy density environments tend to be low mass, fast-rotators.
Moreover, cold gas in clusters and fields may have different origins. The star
formation rate increases with cold gas mass for field galaxies but it appears
to be uncorrelated with cold gas for cluster galaxies.
