  The density dependence of the symmetry energy in relativistic mean-field
models with density dependent couplings is discussed in terms of the possible
opening of nucleonic direct Urca processes inside neutron stars, which induce a
very rapid cooling of the star. The modification of the parametrization of the
isospin channel of two models, DD2 and DDMEX, keeping the same isoscalar
properties is considered and the implications are discussed. Within the models
discussed it is not possible the onset of nucleonic direct Urca processes in
stars with a mass below $\sim1.6-1.8\, M_\odot$. The lowest masses that allow
direct Urca processes are associated to a slope of the symmetry energy $\sim60$
MeV and a symmetry energy incompressibility close to zero. It is shown that the
parametrization of the isospin channel proposed destroys the correlation
between symmetry energy slope and incompressibility previously identified in
several works.
