There are many possible matter models in General Relativity. In Cosmology, a family of simple models are obtained by using perfect fluids and the assumption that $p = w \rho$ for some $w$ that in simple models is taken to be a constant. In this framework, notice your model consists of taking $w = 0$.
This is actually a fairly common matter model, which we often refer to as "dust" or "cold matter". It is very well studied and you can find details about it on most textbooks on General Relativity and/or Cosmology. See, e.g., Sec. 8.3 onward of Sean Carroll's, Sec. 5.3 of Barbara Ryden's, Chap. 5 of Wald's, and so on. Notice that the accelerated expansion of the Universe was discovered in the late 90's, so older books (such as Wald's) are a bit outdated.
Changing the matter models leads to many changes in the geometry. For example, matter-filled Universes might recollapse into a Big Crunch. This might depend on the spatial curvature of the Universe, as discussed on the references I mentioned, but I recall that this does hold if the Universe is spatially a $3$-sphere, which implies the Universe being finite.
In other words, yes, at least in some cases the positive energy of everything leads the Universe to a recollapse, pulling everything together back into a singularity. In some cases (I'm not sure whether in all of them) this happens just like a reverse Big Bang: the Universe reaches a maximum size and then, in large scale, things behave just as if you were rewinding the tape, as you see the Universe shrink again and collapse onto itself.
Notice, however, that $w = 0$ is not a vacuum model. It is well-suited to describe matter with no pressure, which we usually call dust, but there is no motivation (at least that I know of) to treat it as if it was vacuum. The cosmological models I'm mentioning do assume though that there is a fluid filling the entire Universe with $\rho > 0$ and $p = 0$, but this is taken to be an approximation (just like water is made of molecules but we treat it as a continuous fluid, the Universe is filled with galaxies but in large scale we treat it as a continuous fluid).