  Both the symmetry energy part and excluded volume corrections to the equation
of state play an important role for the neutron star interior structure and
composition, namely for the profile of the baryon density and the proton
fraction. While the symmetry energy uniquely determines the proton fraction,
excluded volume effects control the maximum density values inside neutron
stars. Observations of cooling neutron stars indicate that the fast direct Urca
cooling is not operative for the typical, low mass stars, pointing at proton
fractions that lie below the threshold for the onset of direct Urca cooling
process. This in turn, restricts the density range admissible in neutron star
interiors and may require an excluded volume correction. In this contribution
we discuss the interplay between fast cooling, symmetry energy and excluded
volume corrections to the equation of state that would be required to fulfil
the direct Urca cooling constraint.
