It is conserved, except at cosmological scales. Locally GR conserves energy the same as any other self-consistent physical theory.
That's incorrect. Energy conservation can be violated at a much smaller scale, e.g. when gravitational waves are involved, or redshift phenomena (e.g. in Schwarzschild).
Yes, we usually talk about the fact that gravitational waves can carry energy but what exactly is their energy content? And what exactly is the conservation equation here?
> Locally GR conserves energy the same as any other self-consistent physical theory.
Locally, you can write down a divergence equation for the energy momentum tensor, yes. However, locally we're in Minkowski space anyway, so that is not really surprising.
The point is that the local divergence equation doesn't take into account the energy carried by the gravitational field itself. To give another example beyond gravitational waves: A Schwarzschild black hole carries mass, even though it is a vacuum solution to the Einstein field equations.
See also https://en.m.wikipedia.org/wiki/Mass_in_general_relativity and in particular the section on quasi-local mass.
That would be great if energy were a conserved quantity in General Relativity, which it isn't. Heck, we don't even know how to write down the total energy/momentum of a given spacetime volume.