  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.
