The vast high entropy alloy (HEA) composition space is promising for discovery of new material phases with unique properties. This study explores the potential to achieve rare-earth-free high magnetic anisotropy materials in single-phase HEA thin films. Thin films of FeCoNiMnCu sputtered on thermally oxidized Si/SiO2 substrates at room temperature are magnetically soft, with a coercivity on the order of 10 Oe. After post-deposition rapid thermal annealing (RTA), the films exhibit a single face-centered-cubic phase, with an almost 40-fold increase in coercivity. Inclusion of 50 at.% Pt in the film leads to ordering of a single L10 high entropy intermetallic phase after RTA, along with high magnetic anisotropy and 3 orders of magnitude coercivity increase. These results demonstrate a promising HEA approach to achieve high magnetic anisotropy materials using RTA.