Using the cosmic virial theorem, Press-Schechter analysis, and numerical simulations, we compute the expected X-ray background (XRB) from the diffuse intergalactic medium (IGM) with the clumping factor expected from gravitational shock heating. The predicted fluxes and temperatures are ruled out by the observed XRB. The predicted clumping can be reduced by entropy injection. The required energy is computed from the two-point correlation function, as well as from Press-Schechter formalisms. The minimal energy injection of I keV per nucleon excludes radiative or gravitational heating as a primary energy source. We argue that the IGM must have been heated through violent mechanical processes such as massive supernova bursts. If the heating proceeded through supernova explosions, it likely proceeded in bursts which may be observable in high-redshift supernova searches. Within our model, we reproduce the observed cluster luminosity-temperature relation with energy injection of 1 keV per nucleon if this injection is assumed to be uncorrelated with the local density. These parameters predict that the diffuse IGM soft XRB has a temperature of ～1 keV with a flux near 10 keV cm~(-2) s~(-1) sr~(-1) keV~(-1), which may be detectable in the near future.
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