I'm currently thinking about a starmap in the background, because some players reported that the black background is boring after a while
Might be nice. But I'd advice to keep it pretty "low key" so it doesn't' get distracting/annoying. The current black background let's you focus on the obstacles whick is the main point of the game. IMHO you don't want to distract the player too much from that. But, if the star background was toned down a lot and something you can visually ignore when you need to, then I guess it could work - just don't make it too prominent, please
Personally I'd much rather have some cool sound effects when hitting those environment changing vertical lines, checkpoints, walls etc.
Another idea would be to add some "pickups" like "double speed" or whatnot - like, you can pick it up and get a super time on the level but it'll be fast/difficult as hell or you can try to stear clear of it if you can't handle it.
Ohh and maybe you should make it so that when a player restarts from a checkpoint they restart from some fixed "known good" position. I've gotten stuck in a "perpetual death loop" quite a few times because where I hit the checkpoint caused me to restart from a position where it was impossible not to die instantly - that's rather annoying.