For early-type galaxies, the ability to sustain a corona of hot, X-ray emitting gas could have played a key role in quenching their star-formation history. Yet, it is still unclear what drives the precise amount of hot gas around these galaxies. By combining photometric and spectroscopic measurements for the early-type galaxies observed during the Atlas3D integral-field survey with measurements of their X-ray luminosity based on X-ray data of both low and high spatial resolution we conclude that the hot-gas content of early-type galaxies can depend on their dynamical structure. Specifically, whereas slow rotators generally have X-ray halos with luminosity L_X,gas and temperature T values that are in line with what is expected if the hot-gas emission is sustained by the thermalisaton of the kinetic energy carried by the stellar-mass loss material, fast rotators tend to display L_X,gas values that fall consistently below the prediction of this model, with similar T values that do not scale with the stellar kinetic energy as observed in the case of slow rotators. Considering that fast rotators are likely to be intrinsically flatter than slow rotators, and that the few L_X,gas-deficient slow rotators also happen to be relatively flat, the observed L_X,gas deficiency in these objects would support the hypothesis whereby flatter galaxies have a harder time in retaining their hot gas. We discuss the implications that a different hot-gas content could have on the fate of both acquired and internally-produced gaseous material, considering in particular how the L_X,gas deficiency of fast rotators would make them more capable to recycle the stellar-mass loss material into new stars than slow rotators. This is consistent with the finding that molecular gas and young stars are detected only in fast rotators in the Atlas3D sample, and that fast rotators tend to dustier than slow rotators. [Abridged]
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