  We relate the star formation from cold baryons in virialized structures to
the X-ray properties of the associated diffuse, hot baryonic component. Our
computations use the standard ``semi-analytic'' models to describe i) the
evolution of dark matter halos through merging after the hierarchical
clustering, ii) the star formation governed by radiative cooling and by
supernova feedback, iii) the hydro- and thermodynamics of the hot gas, rendered
with our Punctuated Equilibria model. So we relate the X-ray observables
concerning the intra-cluster medium to the thermal energy of the gas pre-heated
and expelled by supernovae following star formation, and then accreted during
the subsequent merging events. We show that at fluxes fainter than $F_X\approx
10^{-15}$ erg/cm$^2 $ s (well within the reach of next generation X-ray
observatories) the X-ray counts of extended extragalactic sources (as well as
the faint end of the luminosity function, the contribution to the soft X-ray
background, and the $L_X-T$ correlation at the group scales) increase
considerably when the star formation rate is enhanced for z>1 as indicated by
growing optical/infrared evidence. Specifically, the counts in the range 0.5-2
keV are increased by factors $\sim 4$ when the the feedback is decreased and
star formation is enhanced as to yield a flat shape of the star formation rate
for 2<z<4.
