We studied nonlinear magnetic anisotropy changes to the DC bias voltage of magnetic tunnel junctions (MTJs) with capping layers of different thermal resistances. We found that increasing the thickness of MgO capping layers (in the range 0.3-0.5 nm) in MTJs enhances the Joule heating-induced magnetic anisotropy change, which indicates an enhancement of the interfacial thermal resistance at the FeB|MgO capping layer interface. This enhanced interfacial thermal resistance may be attributed to roughness at the FeB|MgO interface. Moreover, we observed a larger power-driven magnetic anisotropy change of 3.21 µJ W-1m-1 in the MTJ with a composite MgO (0.3 nm)|W (2 nm)|MgO (0.4 nm) capping layer. This research supports methods of efficient spin manipulation of spintronic devices such as microwave devices and magnetic memories.