I don't have much to add that would be helpful, but it seems to be
correct that there is no quantity known in General Relativity that
globally has the properties of a total conserved energy. I heard Roger
Penrose give a talk at NYU a few years back where he had proposed a
particular quantity that might play such a role around a black hole (if
my memory serves -- and it may not).
I am very far from being an expert on this, and I can't add any more,
but I'll ask a colleague or two.
J. Epstein