Can you make precise what exactly you mean by “volume” here? Is it a 3-volume of space, as in a geodesic ball for example? Or a 4-volume of spacetime?
Both of these depend on the metric, as do all measurements of lengths, angles, areas, and n-volumes in general on a differentiable manifold.
For 3-volumes this is immediately obvious without any further ado, since FLRW spacetime is not a vacuum solution, and thus the Ricci tensor does not vanish. Therefore 3-volumes are not, in general, conserved as they age into the future, irrespective of your choice of coordinates. The only way to get them to be remain constant would be to choose a cosmological constant of just the right value so that you end up with a constant scale factor. But that is not compatible with observations.
For 4-volumes of spacetimes, the determinant of the metric enters as part of the volume element, so when you perform the integration to find the total volume of a given spacetime region, the result will explicitly depend on the times you use as integration limits, since the scale factor will be in there. So again, expansion has an effect here.
All of these are general mathematical results in differential geometry, and not specific to just GR as a theory.
I’m happy to show you the mathematical expressions if you need to see them (but I’m sure you’ve seen enough of me around here to know that I’m not just making this up). Otherwise Misner/Thorne/Wheeler has a good overview on how to construct general volumes on differentiable manifolds, or you can take a quick look on Wiki as well.