  We study the thermal evolution of neutron stars described within the equation
of state with induced surface tension (IST) that reproduces properties of
normal nuclear matter, fulfills the proton flow constraint, provides a
high-quality description of hadron multiplicities created during the
nuclear-nuclear collision experiments, and it is equally compatible with the
constraints from astrophysical observations and the GW170817 event. The model
features strong direct Urca processes for the stars above $1.91~M_{\odot}$. The
IST equation of state shows very good agreement with the available cooling
data, even without introducing nuclear pairing. We also analysed the effect of
the singlet proton/neutron and triplet neutron pairing on the cooling of
neutron stars of different mass. We show that the description of the compact
object in the center of the Cassiopeia A does not necessarily require an
inclusion of neutron superfluidity and/or proton superconductivity. Our results
indicate that data of Cassiopeia A can be adequately well reproduced by a
$1.66~M_{\odot}$ star with an atmosphere of light elements. Moreover, the IST
EoS reproduces each of the observational datasets for the surface temperature
of Cassiopeia A either by a rapidly cooling $\sim$ $1.955~M_{\odot}$ star with
paired and unpaired matter or by a $1.91~M_{\odot}$ star with the inclusion of
neutron and proton pairings in the singlet channel.
