  Energy diffusion due to spontaneous localization (SL) for a
relativistically-fast moving particle is examined. SL is an alternative to
standard quantum theory in which quantum state reduction is treated as a random
physical process which is incorporated into the Schr\"odinger equation in an
observer-independent way. These models make predictions in conflict with
standard quantum theory one of which is non conservation of energy. On the
basis of proposed relativistic extensions of SL it is argued that for a single
localized particle, non-relativistic SL should remain valid in the rest frame
of the particle. The implication is that relativistic calculations can be
performed by transforming non-relativistic results from the particle rest frame
to the frame of an inertial observer. This is demonstrated by considering a
relativistic stream of non-interacting particles of cosmological origin and
showing how their energy distribution evolves as a result of SL as they
traverse the Universe. A solution is presented and the potential for
astrophysical observations is discussed.
