Total energy of the universe

In fact, two very well-known mathematicians, Schoen and Yau, in a much quoted paper, proved the long standing conjecture that the ADM mass is always POSITIVE (except for flat space). Here is the reference and abstract:

Commun. math. Phys. 65, 45--76 (1979)
On the Proof of the Positive Mass Conjecture in General Relativity
Richard Schoen and Shing-Tung Yau
Abstract. Let M be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass of M as viewed from spatial infinity (the ADM mass) must be positive unless M is the fiat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non- negative with equality only if the space-time :is fiat.

In the context of general relativity, the universe is described as a lorentzian spacetime subject to a coupled system of PDEs konwn as the Einstein field equations. These relate the curvature of the spacetime to a tensor field depending on the matter distribution. More precisely, the Einstein field equations say that in some units the Einstein tensor of the spacetime equals the energy-momentum tensor of the matter distribution. The claim you have come across is probably just a paraphrase of the Einstein field equations.

The truth lies elsewhere, as usual. In a gravitational theory energy is tricky to define. The reason is that it is not a function, but simply a component of a tensor and as such it does not have an invariant meaning and depends on a choice of coordinates.

The way relativists get around this problem (in some cases) is the so-called ADM energy. Essentially for spacetimes with "nice" asymptotics, by which one means spacetimes which are asymptotically flat, or more generally asymptotically to a spacetime of constant sectional curvature, one can define a notion of energy (often called mass) by subtracting off the energy of the asymptotic geometry.

The wikipedia article on mass in general relativity goes into more detail and has some links.

When people claim that total energy is zero in a gravitational theory, they usually mean something kind of trivial. In an ordinary field theory, we can define the stress-energy tensor by varying the Lagrangian with respect to the metric. But in a gravitational theory, the metric is a dynamical field, so this variation just gives an equation of motion for the metric that should be zero on a classical solution.

What I've just said sounds fairly classical; the Wheeler-DeWitt equation is a version of this idea in the quantum context.

At low energies, where gravity is weak, you can of course just talk about the energy of a configuration of matter, independent of gravity, but this will not be conserved in general relativity. (A cosmological constant is a standard example where you can see that it's hard to make sense of energy conservation in the presence of gravity.)

As the others have said, the ADM energy is sometimes a more useful definition of energy in GR.