If black holes turn out to b a gravitational condensate with no event horizon, you could have FTL and a hyperbolic space time at the same time, since the condensate would have negative energy pressure.

I've actually been wondering what the simplest experimental proof of relativity would be though. I know GPS is one practical application that has to account for the way gravity bends time which is based on light having a constant speed, but that's hard to demonstrate.