Yes. There’s no reason it can’t move faster than light.
Gravitational waves propagate at $c$ because of how spacetime geometry works - if you interpret the wave equation in the way you need to, you get a speed of $c$ for it. But spacetime curvature in general can travel at any speed.
What you are not allowed to do is send objects along spacelike curves. Things in general (photons, protons, etc) are not allowed to do that (in fact nothing really is) because if you travel along spacelike curves you’re going to a) experience imaginary proper time (bad) and b) be able to back up into your own past and screw with causality. Very bad. However space can be curved so that I can go at any speed without getting onto a spacelike curve (as Paulina mentions, this is the basis for the Alcubierre drive, among others). In flat spacetime, any curve going faster than $c$ is spacelike, but in Alcubierre spacetime within the warp bubble, most curves with any speed are timelike, meaning that I as a massive human being can travel them safely.
Note that the Alcubierre drive doesn't actually let you go faster than a photon - it just speeds up photons, so the speed limit is increased. This is the case with other warp drives as well.
So it’s easy to get curvature to go faster-than-light. Sometimes that curvature requires negative energy, which we can’t get to, but there are contrived circumstances where curvature allows for faster-than-$c$ speeds. Most of the time and especially in flat space curvature changes at around $c$ naturally, though.