In the standard Friedmann cosmology the black-body radiation spectrum is usually taken (without explicit proof as far as we know) to have the same familiarT 4-form that it has in a flat space. With explicit use of the equation of motion of a quantized massless field propagating in a curved background Robertson-Walker metric we show (for the readily tractable scalar field case) that the assumption is in fact true for an open Universe. For a closed Universe, we find that there is an in principle modification to theT 4-law. Unfortunately, the correction turns out to be too small to be experimentally detectable. In passing, we also obtain a simple derivation for the cosmological red shift of frequencies.