In Newtonian mechanics, a particle might gain kinetic energy while a corresponding gravitational potential energy decreases, thus you get that kind of conservation of energy. The total energy is the same before and after any event. However, the amount of energy depends on who's looking.
In Special Relativity a transfer of energy has to happen at an event (a specific time and specific location), so you have to have changes in kinetic energy be compensated by a loss of energy in another body or field. An example is the electromagnetic field which has an energy density, momentum density, and stress at every point in spacetime. Energy can transfer from the electromagnetic field to the particle and thus you get conservation of energy. It can be expressed by saying the energy in some region of space at some time is equal to the energy at an earlier time plus or minus the net flux of energy in or out of that region of space during that time interval. But energy conservation is just one part of a unified energy-momentum conservation, and that conservation can be expressed in a frame independent manner.
In General Relativity you generalize the kind of conservation of energy as is found in Special Relativity, but the tensor T, called the stress-energy tensor, that keeps track of the energy density, momentum density and stress at every event in spacetime is actually the same tensor from Special Relativity, and so it has no terms that correspond to gravitational potential energy. Breaking the stress-energy tensor into just an energy part is frame dependent and General Relativity is formulated in a frame independent manner. Some people try to make an energy psuedo-tensor, but that is a different tensor. And it is the stress-energy tensor T (not the psuedo-tensor) that is the source of the gravitational curvature, just as charges and currents are the source of electromagnetic waves.
So simply put, don't expect General Relativity to have something like "total energy of the universe", because that's just something that isn't naturally there. There is a stress-energy tensor, which if you pick a frame gives you an energy density at an event, but there is usually no natural frame, so no natural energy density.
But when talking about the stress-energy tensor in different epochs, there might be a sense where is has certain properties and at other times has other properties. One property a stress-energy tensor can have is whether it satisfies various so-called energy conditions. And a common consequence of many energy conditions is that the energy density (for every frame) is non-negative. So the question about whether a particular stress-energy tensor has a negative energy density is a legitimate question in General Relativity.