  We search for possible correlations between neutron star observables and thermodynamic quantities that characterize high density nuclear matter. We generate a set of model-independent equations of state describing stellar matter from a Taylor expansion around saturation density. Each equation of state which is a functional of the nuclear matter parameters is thermodynamically consistent, causal and compatible with astrophysical observations. We find that the neutron star tidal deformability and radius are strongly correlated with the pressure, the energy density and the sound velocity at different densities. Similar correlations are also exhibited by a large set of mean-field models based on non-relativistic and relativistic nuclear energy density functionals. These model independent correlations can be employed to constrain the equation of state at different densities above saturation from measurements of NS properties with multi-messenger observations. In particular, precise constraints on the radius of PSR J0030+0451 thanks to NICER observations would allow to better infer the properties of matter around two times the nuclear saturation density. 
Empirical constraints on the high-density equation of state from   multi-messenger observables