  Hot intergalactic gas in clusters, groups, and filaments emanates a
continuous background of 0.5-2.0 keV X-rays that ought to be detectable with
the new generation of X-ray observatories. Here we present selected results
from a program to simulate the surface-brightness distribution of this
background with an adaptive-mesh cosmological hydrodynamics code. We show that
the bright end of this distribution is well approximated by combining the
cluster temperature function with a beta-model for surface brightness and
appropriate luminosity-temperature and core radius-luminosity relations. Our
simulations verify that the X-ray background from hot gas vastly exceeds
observational limits if non-gravitational processes do not modify the
intergalactic entropy distribution. An entropy floor of ~100 keV cm^2, which
could be established by either heating or cooling, appears necessary to
reconcile the simulated background with observations. Because the X-ray
background distribution is so sensitive to the effects of non-gravitational
processes, it offers a way to constrain the thermal history of the
intergalactic medium that is independent of the uncertainties associated with
surveys of clusters and groups.
