I am building a laundry room with a closet. I could frame the walls and then add backer board. However, it would be easier to apply backer board to the entire room and then frame the closet on top of the backer board. Is this an acceptable method?
You are trying to avoid a few backer board cuts related to the closet area. However, you are also increasing the height of the floor. If your existing walls are the standard 8' height, you will have to cut every stud of the new walls in order to frame them correctly. You are just creating more work for yourself. Either way you go, there will be an extra step involved and no significant time would be saved. Frame the walls first, then lay the flooring.
I'll play devil's advocate and state that I'd rather cut a few studs off than cut and fit and fasten more pieces of cementboard around putzy closet walls any day. It's a net time saver. There are no structural concerns with framing on top, and the floor will end up more rigid, having fewer seams.
Also, since they're non-bearing interior walls, a person could use 96" studs (cut to length), as opposed to pre-cut studs, and single top plates.