  During very early age of neutron stars, the core cools down faster compared
to the crust creating a large thermal gradient in the interior of the star.
During $10-100$ years, a cooling wave propagates from the core to the crust
causing the interior of the star to thermalize. During this duration thermal
properties of the core material is of great importance to understand the
dynamics of the interior of the star. The heat capacity and thermal
conductivity of the core depends on the behaviour of matter inside the core. We
investigate these two properties in case of magnetars. Due to presence of large
magnetic field, the proton superconductivity is quenched partially inside the
magnetars depending upon the comparative values of upper critical field and the
strength of the magnetic field present. This produces non-uniformity in the
behaviour of matter throughout the star. Moreover, such non-uniformity arises
from the variation of nature of the pairing and values of the pairing gap
energy. We find that the heat capacity is substantially reduced due to the
presence of superfluidity. On the other hand, the thermal conductivity of
neutron is enhanced due to proton superconductivity and gets reduced due to
neutron superfluidity. Hence, the variation of the thermal properties due to
superfluidity in presence of magnetic field is different at different radius
inside the star. However, in all the cases the %minimum maximum variation is of
the order one. This affects the thermal relaxation time of the star and
eventually its the thermal evolution.
