I assume you are making a distinction between "energy pressure" and "pressure energy". If not, there is a vast amount of research to do; if so, can you point in an appropriate direction?
Nice to see someone thinking on this forum so lets first do the standard science.
So the standard GR version basically has at the bottom of it the concept of dark energy (this is the hole that Bill and Rede fell down), you can't believe in GR but not in dark energy they are joined at the hip
There are two ways that it is conceptually taught as either a cosmological constant or as quintessence depending on the scientist you ask and what they believe/promote
The cosmological constant version goes that there is a cost to energize a portion of space. Why the cost exists involves a bit of hand waving and saying that is just the way it is. The key criteria it sets out are
1.) The vacuum energy density is constant because there is nothing for it to depend on. This fits back to the original premise of uniformity of all space.
2.) Using classical thermodynamics and assuming the constant exists and you consider the volume of space is finite. A change in volume (dV) requires work done equal to a change of energy E = -P*dV, where P is the pressure, E being energy. However as your space volume has increased and you assumed a cosmological constant for space then the total energy must have increased. So the conclusion is that P must in fact be negative to give you a positive energy value for (E) ... We don't allow negative energy in classic physics !!!!!
Now there are a number of problems in all that, some you will immediately see others are not so obvious. One important not so obvious problems is the cosmic constant is incompatible with the standard model at low matter denisities, how problematic this is depends on your faith in the standard model.
If the cosmological constant is your thing then the favoured model is called Lambda-CDM.
Your second choice quintessence conjectures a fifth unknown scalar force which is varying slightly over time. Paul J. Steinhard's cornerstone work on the idea is technical but should be somewhat understandable by even a layman http://physics.princeton.edu/~steinh/steinhardt.pdf
Scalar fields are predicted by the standard model and string theory but there are problems with renormalization of a slow changing scalar field. Which set of problems is worse well that is in the eye of the beholder.
If you get excited by quintessence you may care to look at the most favoured model at the moment called slow rolling inflation.
So how do you distinguish between the two ... well in short detailed and prolonged measurement of the fundemental forces