Is a Gravastar more dense than a Neutron Star? Given:  


*

*A Neutron star is the collapsed core of a larger star;

*Models for Neutron stars suggest they are composed almost entirely of neutrons;


And if:


*

*Bose-Einstein condensates are extremely dense agglomerations of near zero entropy bosons



Wouldn't a Gravastar be more dense than a Neutron star since it has more mass-density $\rho$ than a Neutron star, which has more entropy?

 A: 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. 
See https://arxiv.org/abs/0706.1513
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 
https://phys.org/news/2016-10-ligo-black-holes-gravastars.html
So, gravastars are still a low probability resolution of the black hole mysteries we still need to resolve. 
