Regular black holes without mass-inflation instability and gravastars from modified gravity

We derive regular black-hole solutions, including the Hayward metric, from four-dimensional action principles involving vector fields in addition to the metric. These black holes possess additional hair associated with the vector fields, manifesting as free integration constants that regularize the geometry. These constants can be chosen such that regular black holes of all masses are extremal. As a result, they have vanishing surface gravity and are not susceptible to mass-inflation instability. We also discover another regular black-hole metric with these properties, which constitutes a gravastar for an appropriate choice of integration constant.