From our results we can conclude the following:
- The constrained Monte Carlo method is capable to evaluate temperature-dependent properties. We have shown its novel capability to evaluate the temperature dependence of the magnetic anisotropy, which is an important quantity for technological applications in hard magnetic materials. In the case of bulk materials we recovered numerically the analytical scaling law of Callen and Callen.
- We have also shown that the method enables the separation of the temperature dependent surface anisotropy as an intrinsic system parameter, independent of the thin film thickness. Our results demonstrate a linear temperature dependence of the surface anisotropy, consistent with the experimental results in Gd [133,134], Ni  and Fe  grown on different substrates.
- We have also shown that the competition between the surface and the bulk anisotropies could produce a re-orientation of the effective easy axis. The re-orientation temperature depends on the system size.
- We have investigated the scaling of the anisotropy in thin films with magnetization. In ultrathin films without bulk moments, the surface anisotropy scales with the surface magnetization following the Callen-Callen law as corresponds to the uniaxial case. In other cases with both surface and bulk anisotropy contributions we report no universal scaling behavior of the effective anisotropy.