  We examine the cosmological implications of space-time non-commutativity,
discovering yet another realization of the varying speed of light model.
  Our starting point is the well-known fact that non-commutativity leads to
deformed dispersion relations, relating energy and momentum, implying a
frequency dependent speed of light. A Hot Big Bang Universe therefore
experiences a higher speed of light as it gets hotter. We study the statistical
physics of this "deformed radiation", recovering standard results at low
temperatures, but a number of novelties at high temperatures: a deformed
Planck's spectrum, a temperature dependent equation of state $w=p/\rho$
(ranging from 1/3 to infinity), a new Stephan-Boltzmann law, and a new entropy
relation. These new photon properties closely mimic those of phonons in
crystals, hardly a surprising analogy. They combine to solve the horizon and
flatness problems, explaining also the large entropy of the Universe. We also
show how one would find a direct imprint of non-commutativity in the spectrum
of a cosmic graviton background, should it ever be detected.
