  We study thermal states of transiently accreting neutron stars (with mean
accretion rates $\dot{M} \sim 10^{-14}-10^{-9}$ M$_\odot$ yr$^{-1}$) determined
by the deep crustal heating of accreted matter sinking into stellar interiors.
We formalize a direct correspondence of this problem to the problem of cooling
neutron stars. Using a simple toy model we analyze the most important factors
which affect the thermal states of accreting stars: a strong superfluidity in
the cores of low-mass stars and a fast neutrino emission (in nucleon,
pion-condensed, kaon-condensed, or quark phases of dense matter) in the cores
of high-mass stars. We briefly compare the results with the observations of
soft X-ray transients in quiescence. If the upper limit on the quiescent
thermal luminosity of the neutron star in SAX J1808.4-3658 (Campana et al.
2002) is associated with the deep crustal heating, it favors the model of
nucleon neutron-star cores with switched-on direct Urca process.
