  This paper presents the multi-scale temperature structures in the Milky Way
(MW) hot gas, as part of the XMM-Newton Line Emission Analysis Program
(X-LEAP), surveying the O VII, O VIII, and Fe-L band emission features in the
XMM-Newton archive. In particular, we define two temperature tracers, $I_{\rm
OVIII}/I_{\rm OVII}$ (O87) and $I_{\rm FeL}/(I_{\rm OVII}+I_{\rm OVIII})$
(FeO). These two ratios cannot be explained simultaneously using
single-temperature collisional ionization models, which indicates the need for
multi-temperature structures in hot gas. In addition, we show three large-scale
features in the hot gas: the eROSITA bubbles around the Galactic center (GC);
the disk; and the halo. In the eROSITA bubbles, the observed line ratios can be
explained by a log-normal temperature distribution with a median of $\log
T/{\rm K} \approx 6.4$ and a scatter of $\sigma_T \approx 0.2$ dex. Beyond the
bubbles, the line ratio dependence on the Galactic latitude suggests higher
temperatures around the midplane of the MW disk. The scale height of the
temperature variation is estimated to be $\approx$2 kpc assuming an average
distance of $5$ kpc for the hot gas. The halo component is characterized by the
dependence on the distance to the GC, showing a temperature decline from
$\log\,T/{\rm K}\,\approx\, 6.3$ to $5.8$. Furthermore, we extract the
auto-correlation and cross-correlation functions to investigate the small-scale
structures. O87 and FeO ratios show a consistent auto-correlation scale of
$\approx$$ 5^\circ$ (i.e., $\approx$$ 400$ pc at 5 kpc), which is consistent
with expected physical sizes of X-ray bubbles associated with star-forming
regions or supernova remnants. Finally, we examine the cross-correlation
between the hot and UV-detected warm gas, and show an intriguing
anti-correlation.
