Some people say that dark-energy density remains constant in an expanding universe. However this is something that's still rather hypothetical. It's rather atypical too, in that we have no examples of non-conservation of energy. If you move towards the $E=hf$ photon it appears blue-shifted, but it hasn't actually gained any energy. It hasn't changed at all, instead, you changed. If you drop a brick, gravity converts potential energy into kinetic energy. But it doesn't create any energy. It doesn't do work on the brick, instead you do work on the brick when you lift it up. In similar vein the ascending brick doesn't lose any energy. Nor does the ascending photon. See Einstein talking about red-shift in the digital papers hosted at Princeton. Note that he doesn't say light changes frequency as it climbs out of the gravitational field. He says it's emitted at a lower frequency. There are no perpetual motion machines.
However some people say things like "any reasonable definition of energy that has been found leads to non-conservation in many GR scenarios". Only I don't know of any example of non-conservation of energy. Can you provide any? Either in a GR scenario, or in any other scenario?
I will award a 200-point bounty on this question. I will award it to the answer that I think is the best. If I don't much like any of the answers, I will still award the bounty to one of them.
Edit: now the question has been closed I have to withdraw the offer of a bounty I'm afraid. Apologies to those who have answered.
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