Depends on the cosmos.

Below energy means any nonzero of any component of the stress-energy tensor T.

In vacuum flat spacetime, Tmunu def. 0, so the energy at every point is 0, the total energy is 0, and the average energy density at every point is 0.

We can take this logic to several families of vacuum spacetime.

In the sort of steady-state universe discussed in the early 20th century there is some distribution of stress-energy, varying the energy at any given point, but total energy = const., average energy density = const.

The problem is when we start adjusting the number of points without altering the stress-energy. Interiors of (e.g. Lemaître-Tolman-Bondi) black holes develop lots of new spacetime points, but most of those points have zero stress-energy. Futures of expanding universes have lots of new spacetime points, and most of them have zero stress-energy. Put the two of them together in a "swiss cheese" spacetime, (we can cast a clumping \Lambda-CDM cosmos as one of those, with galaxy clusters forming and ultimately collapsing into ginormous black holes, and extragalactic space evolving into exceptionally hard vacuum "cheese" surrounding these holes), and you have a recipe for a growing number of points in space which with no stress-energy in them. For a sufficiently long-lived expanding swiss-cheese universe, what's the average value in the components of the stress-energy tensor? Zero. Approaching future timelike infinity, good luck finding any non-zeros at all.

Weiss & Baez have a nice but brief expansion upon the \nabla_mu \cdot T^munu = 0 point that Carroll at your link declined to explain :) under the "Divergence and Integration" heading at https://math.ucr.edu/home//baez/physics/Relativity/GR/energy...

Like them I would not go into Noether's (first) theorem in any detail in explaining this even to an audience with more familiarity with physical cosmology than Carroll had in mind, for the following reason.

Note that while Weiss & Baez write that certain cases meet most of the conditions required for Noether's theorem to hold, they do not say with specificity that in general dynamical spacetimes do not possess the conditions required by Noether's theorem, as noted in her commentary about the non-independence of the Euler-Lagrange equations in her second theorem paper E. Noether, Nachr. d. Konig. Gesellsch. d. Wiss. zu Gottingen, Math-phys. Klasse pp. 235–257 (1918). In modern language the current j^mu_beta vanishes iff the divergence of \sqrt(-det g_munu)\cdot T^mu_beta = 0. I can dig up some refs for this if you're realllllllllly keen, although I'd probably have to start with parts of Katherine Brading & Harvey Brown's 2000s-era "really, which symmetries?" work.

I'd rather just say, hey, in reasonably realistic spacetimes we break all sorts of symmetries compared to popular classics, and even those broke one or more symmetries compared to flat spacetime, and among the consequences are that things you (in flat spacetime) think are scalars (e.g. energy) may demand to be treated as tensor contractions, and vectors (e.g. momentum) are often even worse. These quantities should take on different names to avoid confusion. And those are what you are conserving, approximately conserving, or not conserving.

Or, to the point, any practical definition of energy is spacetime-dependent and the spacetime of special relativity is just one of an infinite number of possible spacetimes, and the only one that in 3+1 dimensions is uncurved.

The bright side is that in Lorentzian (3+1) spacetimes, you always have at least an infinitesimal patch of flat spacetime around any point, and often an even larger patch of effectively flat spacetime, so we have still have the local conservation laws ancestors discovered.

In most parts of our universe, and everywhere on Earth, the radius of (Riemann tensor) curvature is large. So one can talk with virtually perfect accuracy to many digits of precision about special-relativistic energies or momenta of particle colliding experiments (including natural "experiments" carried out in post-supernova/post-white dwarf deflagration nebula), and use conservation laws to decide there must be some species of particle carrying away some seemingly-missing energy or momentum, even though one can use atomic clocks or Pound-Rebka devices to measure the actual nonflatness of the patch of spacetime the collider building is in, or observe with spectroscopy and interferometry manifestly general-relativistic effects around the stellar remnants. All of that is relevant if one is hunting axions, for example, where one will benefit from local Lorentz invariance and the conservation laws that flow from that.

This is a little important in the face of people misunderstanding "energy is not conserved in general relativity" as permitting EmDrive-type nonsense.

And finally, recalling my first line, in classical General Relativity we can define cosmologies for curved spacetimes with arbitrary integer numbers of dimensions greater than 1 such that non-conservative extra functions (encoding a "friction" dissipative coupling between a metric and stress-energy, for instance; people have actually done this in bi-metric gravity for inflation reasons, and so have "neo-(quasi-)steady-state-universe" people) must accompany the Lagrangian. This accompaniment is incompatible with Noether's (first) theorem, even locally, except where (and if) the dissipative force has decayed away.