  The recent measurements of the masses of the pulsar J00737-3039B and of the
companion J1756-2251 and pulsars PSR J1614-2230, PSR J0348-0432 demonstrate the
existence of compact stars with masses in a broad range from 1.2 to 2
$M_\odot$. To fulfill the constraint $M_{\rm max}>2M_{\odot}$ and to
demonstrate the possibility of cooling scenarios for purely hadronic and
further for hybrid stars we exploit the stiff DD2 hadronic equation of state
producing a maximum neutron star mass $M\simeq 2.43 M_{\odot}$. We show that
the "nuclear medium cooling" scenario for neutron stars comfortably explains
the whole set of cooling curves just by a variation of the star masses without
the necessity for the occurrence of the direct Urca reaction. To describe the
cooling data with the very stiff DD2 equation of state we select a proton gap
profile from those exploited in the literature and allow for a variation of the
effective pion gap controlling the efficiency of the medium modified Urca
process. Fast cooling of young neutron stars like it is seen in the data for
Cas A is explained with the DD2 equation of state when the following conditions
are provided: the presence of an efficient medium modified Urca process, and a
large proton gap at densities $n\le 2n_0$ vanishing for $n\ge (2.5 - 3) n_0$,
where $n_0$ is the saturation nuclear density.
