  Hot gaseous atmospheres that permeate galaxies and extend far beyond their
stellar distribution, where they are commonly referred to as the circumgalactic
medium (CGM), imprint important information about feedback processes powered by
the stellar populations of galaxies and their central supermassive black holes
(SMBH). In this work we study the properties of this hot X-ray emitting medium
using the IllustrisTNG cosmological simulations. We analyse their mock X-ray
spectra, obtained from the diffuse and metal-enriched gas in TNG100 and TNG50,
and compare the results with X-ray observations of nearby early-type galaxies.
The simulations reproduce the observed X-ray luminosities ($L_{\rm X}$) and
temperature ($T_{\rm X})$ at small ($<R_{\rm e}$) and intermediate ($<5R_{\rm
e}$) radii reasonably well. We find that the X-ray properties of lower mass
galaxies depend on their star formation rates. In particular, in the magnitude
range where the star-forming and quenched populations overlap, $M_{\rm
K}\sim-24$ $ (M_*\sim10^{10.7}M_\odot)$, we find that the X-ray luminosities of
star-forming galaxies are on average about an order of magnitude higher than
those of their quenched counterparts. We show that this diversity in $L_{\rm
X}$ is a direct manifestation of the quenching mechanism in the simulations,
where the galaxies are quenched due to gas expulsion driven by SMBH kinetic
feedback. The observed dichotomy in $L_{\rm X}$ is thus an important observable
prediction for the SMBH feedback-based quenching mechanisms implemented in
state-of-the-art cosmological simulations. While the current X-ray observations
of star forming galaxies are broadly consistent with the predictions of the
simulations, the observed samples are small and more decisive tests are
expected from the sensitive all-sky X-ray survey with eROSITA.
