  Within the "nuclear medium cooling" scenario of neutron stars all reliably
known temperature - age data, including those of the central compact objects in
the supernova remnants of Cassiopeia A and XMMU-J1732, can be comfortably
explained by a set of cooling curves obtained by variation of the star mass
within the range of typical observed masses. The recent measurements of the
high masses of the pulsars PSR J1614-2230 and PSR J0348-0432 on the one hand,
and of the low masses for PSR J0737-3039B and the companion of PSR J1756-2251
on the other, provide independent proof for the existence of neutron stars with
masses in a broad range from $\sim 1.2$ to 2 $M_\odot$. The values $M>2
M_{\odot}$ call for sufficiently stiff equations of state for neutron star
matter. We investigate the response of the set of neutron star cooling curves
to a stiffening of the nuclear equation of state so that maximum masses of
about $2.4 M_\odot$ would be accessible and to a deconfinement phase transition
from such stiff nuclear matter in the outer core to color superconducting quark
matter in the inner core. Without readjustment of cooling inputs the mass range
required to cover all cooling data for the stiff DD2 equation of state should
include masses of $2.426 M_\odot$ for describing the fast cooling of CasA while
the existence of a quark matter core accelerates the cooling so that CasA
cooling data are described with a hybrid star of mass $1.674 M_\odot$.
