The Chandrasekhar-Friedman-Schutz (CFS) instability consist in enhancement of perturbations in a rotating star by emission of gravitational waves. In absence of dissipation, all rotating stars should be unstable and the emission of gravitational waves should spin down them rapidly. Observations of rapidly rotating neutron stars indicates that it is not a case and dissipative processes suppress the instability in these objects. It allows to constrain properties of dense matter by confronting observations and theoretical models of suppression of CFS instability.
In this talk, I describe the mechanism of the CFS instability, derive the evolution equation for CFS unstable neutron star and describe up-to date observational constraints on the instability windows – the region of temperature and spin frequencies there the instability can be unsuppressed without contradictions with observations. In particular, I discuss theoretically predicted class of neutron stars -- HOt and Fast NonAccreting Rotators (HOFNARs) – which can emit persistent gravitational waves on cosmological timescale. Special attention is devoted to observations of PSR J0952-0607 -- the second fastest known millisecond pulsar, which was recently detected in X-rays, providing strongest constraints to CFS instability in low temperature neutron stars.