curvature in FLRW

Consider a FLRW-universe: does the curvature change with time in an empty, maximally hyperbolic cosmology? Does the curvature increase if the cosmology is decelerating due to a single fluid with \(w>-1/3\)?

The funkiest way to think about this problem is to invoke the Birkhoff-theorem, which states that spherically symmetric vacuum solutions are necessarily static. In fact, FLRW-universes are generally isotropic, and in the case of an empty universe with full curvature vacuum solutions! For that reason, the curvature invariants don’t depend on time; the universe expands with constant \(\dot{a}\) as \(\ddot{a}=0\) as there is nothing to accelerate or decelerate it.