Correct, and the reason is that in GR there is (in general) no time translation invariance. The associated conserved quantity to time translation invariance is precisely what we call "energy". In our universe this manifests as the expansion of spacetime.
More on this: https://www.preposterousuniverse.com/blog/2010/02/22/energy-...
Interestingly I just read about this in a Lawrence M. Krauss book [0] where it was described that David Hilbert insisted on hiring Noether by the university but was overruled by the male majority that didn't like the idea that a female would teach male students. Hilbert commented that "this is university, not a bath-house!" [1].
[0]: I think it was "The Greatest Story Ever Told—So Far"
[1]: I'm writing that from memory so excuse some inaccuracies.
This isn't quite right. Energy isn't conserved in General Relativity. There is a conservation law involving the stress-energy tensor, but as the name implies, it's not energy alone that's being conserved.
But more generally, Noether's theorem, which leads to this conservation law, is probably the most beautiful theorem in physics: https://en.wikipedia.org/wiki/Noether%27s_theorem. Given that, what's amazing is that Emmy Noether only worked on physics for a relatively short time. She was asked by David Hilbert to work on the problem of invariants in General Relativity, came up with one of the most important theorems in physics, and then moved on to other (extremely important) mathematical work.