Yes, they have to be more dense than a neutron star, but effectively the models for gravastars involve a phase transition for the matter at about the time it would reach the horizon, and the spacetime inside turns into a deSitter spacetime.
It provides a solution to the EFE, but it does require one to posit some fairly exotic matter models. There were concerns as to whether the solution is stable to perturbations and the following paper determined yes. But it also determined that the merger of two of them would create gravitational waves which in the ringdown (the end) phase would be somewhat different from black holes.
In what turns out to be worse news for gravastars (as modeled anyway), a Phys Rev D paper in 2016 showed that the LIGO first grav wave detection in 2014 had those ringdown waves inconsistent with them being from gravastars. See a summary at
So, gravastars are still a low probability resolution of the black hole mysteries we still need to resolve.