Forget about the expansion of the universe; you could ask the exact same question for an observer moving away from a source and detecting a different frequency (hence energy) of the light emitted due to "normal" Doppler effect in a non-expanding universe. Energy (on its own) is not an invariant in (special/general) relativity. You have the same situation for measurements of lengths (i.e. Lorentz contraction). Energy is one coordinate of a four-vector whose "length" is the true invariant.
This is similar to a situation where you draw a set of perpendicular axis on a plane and use them to give coordinates (x, y) to a point. You then rotate the set of axis, keeping the point fix. Your question about the energy amounts to asking "what happens to the x-coordinate of that point?".