Magnetars are neutron stars possessing a magnetic field of about $10^{14}-10^{15}$ G at the surface. Thermodynamic properties of neutron star matter, approximated by pure neutron matter, are considered at finite temperature in strong magnetic fields up to $10^{18}$ G which could be relevant for the inner regions of magnetars. In the model with the Skyrme effective interaction, it is shown that a thermodynamically stable branch of solutions for the spin polarization parameter corresponds to the case when the majority of neutron spins are oriented opposite to the direction of the magnetic field (i.e. negative spin polarization). Moreover, starting from some threshold density, the self-consistent equations have also two other branches of solutions, corresponding to positive spin polarization. The influence of finite temperatures on spin polarization remains moderate in the Skyrme model up to temperatures relevant for protoneutron stars. In particular, the scenario with the metastable state characterized by positive spin polarization, considered at zero temperature in Phys. Rev. C 80, 065801 (2009), is preserved at finite temperatures as well. It is shown that, above certain density, the entropy for various branches of spin polarization in neutron matter with the Skyrme interaction in a strong magnetic field shows the unusual behavior, being larger than that of the nonpolarized state. By providing the corresponding low-temperature analysis, we prove that this unexpected behavior should be related to the dependence of the entropy of a spin polarized state on the effective masses of neutrons with spin up and spin down, and to a certain constraint on them which is violated in the respective density range.