Can you change the colour of frosted glass easily?
No, it's not easily changed. I strongly suspect that you'd be better off living with it white, or trading it for a black one, as any paint-based solution will be far inferior to actual colored glass in appearance and durability. 

The best possible paint option might be to paint the bottom black, but it may not give the look you want - however, painting the top black will give you simply the look a black painted surface, not a "frosted glass surface" at all.

You might want to use plasti-dip (or something similar) if experimenting with paint - it can be peeled off most surfaces if you change your mind.

But I'd suggest leaving it alone as being the most likely to succeed.