We discuss energy diffusion due to spontaneous localization (SL) for a relativistically-fast moving particle. Based on evidence from relativistic extensions of SL we argue that non-relativistic SL should remain valid in the particle rest frame. This implies that calculations can be performed by transforming non relativistic results from the particle rest frame to the frame of the observer. We demonstrate this by considering a relativistic stream of non-interacting particles of cosmological origin and showing how their energy distribution evolves as they traverse the Universe. We present a solution and discuss the potential for astrophysical observations.