Epitaxial Fe1-xCox alloy films with x=0.22 and x=0.33 were grown on GaAs(001) by molecular-beam epitaxy in a thickness range of 3 to 80 monolayers (ML). The magnetic properties were investigated by magneto-optic Kerr effect. Ferromagnetic order at room temperature was observed for thicknesses above 4 ML. The in-plane magnetic anisotropy of all films is a superposition of a uniaxial component with the easy axis along [110], which is not discussed here, and a four-fold contribution. The effective uniaxial and four-fold anisotropy constants, K-U(eff) and K-1(eff), were determined by fits to the hard axis magnetization loops. K-1(eff) contains an interface and a volume term which lead to a linear variation of K-1(eff) with the inverse film thickness. It turns out that the surface and volume anisotropy constants are of opposite sign for all alloy compositions causing a sign reversal at a critical thickness, t(crit). This critical thickness seems to be a universal value, which is caused by a general proportionality between the volume and the interface anisotropy constants with the same negative constant of proportionality for Fe1-xCox/GaAs(001), Fe/Au(001), and Fe/Ag(001). This behavior of the four-fold anisotropy constants is consistently explained within Neel's pair energy model for a body-centered-cubic ferromagnet. (C) 2002 American Institute of Physics.