  Magnetic anisotropy plays an essential role in information technology
applications of magnetic materials, providing a means to retain the long-term
stability of a magnetic state in the presence of thermal fluctuations.
Anisotropy consists of a single-ion contribution stemming from the crystal
structure and two-ion terms attributed to the exchange interactions between
magnetic atoms. A lack of robust theory crucially limits the understanding of
the temperature dependence of the anisotropy in pure two-ion and mixed
single-ion and two-ion systems. Here, we use Green's function theory and
atomistic Monte Carlo simulations to determine the temperature scaling of the
effective anisotropy in ferromagnets in these pure and mixed cases, from
saturated to vanishing magnetization. At low temperature, we find that the pure
two-ion anisotropy scales with the reduced magnetization as $k(m) \sim
m^{2.28}$, while the mixed scenario describes the diversity of the temperature
dependence of the anisotropy observed in real materials. The temperature
dependence of the mixed anisotropy determined here is expected to considerably
contribute to the understanding and the control of the thermal properties of
magnetic materials.
