  We explore the implications of a strong first-order phase transition region
in the dense matter equation of state in the interiors of rotating neutron
stars, and the resulting creation of two disjoint families of neutron-star
configurations (the so-called high-mass twins). We numerically obtained
rotating, axisymmetric, and stationary stellar configurations in the framework
of general relativity, and studied their global parameters and stability. The
instability induced by the equation of state divides stable neutron star
configurations into two disjoint families: neutron stars (second family) and
hybrid stars (third family), with an overlapping region in mass, the high-mass
twin-star region. These two regions are divided by an instability strip. Its
existence has interesting astrophysical consequences for rotating neutron
stars. We note that it provides a natural explanation for the rotational
frequency cutoff in the observed distribution of neutron star spins, and for
the apparent lack of back-bending in pulsar timing. It also straightforwardly
enables a substantial energy release in a mini-collapse to another neutron-star
configuration (core quake), or to a black hole.
