I don't see why not. The only problem I can think off is trapped moisture under the slate. It would be nice to lift these up to let the deck have a chance to breath and dry out.
If you notice any excessive mold or algae growth on the deck frame, under the slate, then you will have to remove the slate.
Many good deck stains are very resistant to standing water and you could use a solid color stain around the hot tub. This could be a nice border of color against the other deck stain. Just a thought.