  We study the relative importance of several recent updates of microphysics
input to the neutron star cooling theory and the effects brought about by
superstrong magnetic fields of magnetars, including the effects of the Landau
quantization in their crusts. We use a finite-difference code for simulation of
neutron-star thermal evolution on timescales from hours to megayears with an
updated microphysics input. The consideration of short timescales ($\lesssim1$
yr) is made possible by a treatment of the heat-blanketing envelope without the
quasistationary approximation inherent to its treatment in traditional
neutron-star cooling codes. For the strongly magnetized neutron stars, we take
into account the effects of Landau quantization on thermodynamic functions and
thermal conductivities. We simulate cooling of ordinary neutron stars and
magnetars with non-accreted and accreted crusts and compare the results with
observations. Suppression of radiative and conductive opacities in strongly
quantizing magnetic fields and formation of a condensed radiating surface
substantially enhance the photon luminosity at early ages, making the life of
magnetars brighter but shorter. These effects together with the effect of
strong proton superfluidity, which slows down the cooling of kiloyear-aged
neutron stars, can explain thermal luminosities of about a half of magnetars
without invoking heating mechanisms. Observed thermal luminosities of other
magnetars are still higher than theoretical predictions, which implies heating,
but the effects of quantizing magnetic fields and baryon superfluidity help to
reduce the discrepancy.
