  We discuss the observation that under neutron star conditions of charge
neutrality and $\beta-$equilibrium the contribution from the symmetry energy to
the equation of state (EoS) follows a universal behaviour. We call this
behaviour the conjecture of a Universal Symmetry Energy Contribution (USEC). We
find that an USEC holds provided the density dependence of the symmetry energy
$E_s(n)$ follows a behaviour that limits the proton fraction $x(n)$ to values
below the threshold for the direct Urca (DU) cooling process. The absence of DU
cooling in typical mass neutron stars appears to be supported by the
phenomenology of neutron star cooling data and allows to constrain the
behaviour of $E_s(n)$ at high densities. Two classes of symmetry energy
functions are investigated more in detail to elucidate the USEC. We derive an
analytic formula for the USEC to the neutron star EoS based on the result for
the symmetry energy extracted from isobaric analog states of nuclei.
