In general relativity, you don't actually have a good handle on energy conservation except as a localized phenomenon. Put another way, it is crudely analogous to the energy loss in a Doppler shift. The energy "loss" in regular Doppler shift is simply a consequence of a Lorentz transformation between frames. Imagine then a non-inertial frame. There is no conservation of energy in such a frame.
GR does away with frames altogether.
In the first line of the quoted material, did you mean "special relativity" rather than "general relativity"? It doesn't seem to make sense as it stands, speaking first of frames in GR then the statement the GR gets rid of frames altogether.
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