It's definite that a closed universe has net zero energy. An open universe may be able to avoid it, but not likely.
One aspect is that in curved spacetime, it is difficult to even define what you mean by 'total energy'. The point is that energy is one component of the energy-momentum 4-vector and comparing such vectors at different locations isn't well defined in curved situations.
Think of it like this. Take a tangent vector to a sphere at the equator, pointed in the direction of the equator. Translate it up to the north pole via parallel transport, then back down to the equator along a different longitude line (for specificity, along one 90 degrees from the first). Then translate back along the equator to the original point. The vector that moved via parallel transport will NOT be the same as the original it will have rotated.
Such rotation is inevitable in curved manifolds. In the case of the energy-momentum vector, it means that the energy component, when translated to another point in space will depend on which route you use to do the translation. In other words, to add up the energies from different locations, the answer may well depend on how you compare different points.
So even defining 'total energy' is problematic, to say the least. The only way out is to include an energy component to the curvature itself. And *that* ultimately leads to zero total energy.
