If you have the cosmological constant just right, does there exist a solution to GR representing a stable infinite cubic lattice of point masses $m$ at a distance $d$? How would the stabilizing cosmological constant depend on $m$ and $d$?
I'm betting that the answer changes if you divide $d$ by 2 and divide $m$ by 8, but approaches some limit (for small $d$ and $m$) equal to Einstein's static universe.
The stable lattice would make the overall topology like an unfolding of a 3-torus: each cube is identical to each other, with matching boundary conditions along the borders. Since the 3-torus is a flat solution of the FLRW equations it implies that the cosmological constant must exactly counteract the masses: the negative energy density in a cube volume must equal the mass ($\Lambda \propto m/d^3$).
I doubt there exist any analytic matching condition for lattices of single point masses across the boundary, but certainly the continuum limit makes sense.
This is stable in the sense that as long as nothing moves everything remains in place. However, if you move two masses together homogeneity breaks and I would expect stuff to fall apart quickly (consider the normal models of formation of cosmological voids and filaments). But analyzing it is beyond me, since now we cannot even use matching conditions on the cube boundaries to constrain solutions.
