However, this intergal, which I call PQ for no particularly good reason except to give it a name, is not really the "energy" of the universe. It is not the energy because it depends on one's choice of coordinates, and because it's not necessarily conserved. But it's an interesting quantity, nonetheless.
For standard isotropic cosmologies (FRW cosmologies), PQ is conserved exactly in "dust" universes, and is not exactly conserved in any universe where there is pressure. Our universe is matter dominated at the current time, so the pressure is very low, but it is non-zero, thus PQ is not exactly conserved in our universe, but is approximately conserved. (This assumes that one belives our universe is modelled by a FRW cosmology).
Note that in the past, the universe was radiation dominated rather than matter dominated, making the pressure high, and that in this era of the universe PQ was not even approximately conserved.
This is discussed a bit in MTW, where the above quantity is calculated in more detail, and it is stressed that this quantity should *NOT* be considered to be the "energy of the universe" (even thought it is tempting to do so).