The dependence of critical current density (J_c) on applied magnetic field (B) and angle (ϴ) between B and the c axis has been investigated to determine the anisotropy of MgB₂ thin films at different temperatures. The results have been compared with upper critical field (H_c2) anisotropy (γH) determined by measuring H_c2 for fields applied parallel to the ab plane and the c axis. Contrary to the case of Bi₂Sr₂CaCu₂O₈ and YBa₂Cu₃O_x, where J_c(ϴ,H) scales with H/H_c2(ϴ), J_c(ϴ,H),  for MgB₂ did not follow this scaling. Surprisingly, the temperature dependence of anisotropy determined from J_c(ϴ,H) is similar to the predictions for penetration depth anisotropy of MgB₂. These results have been understood as a combined effect of a two-band theory and the collective pinning model, in which Jc strongly depends on the elastic properties of a vortex lattice.