Inverse optimization of robotic stereotactic lung radiotherapy is typically performed using relatively simple dose calculation algorithm on a single instance of breathing geometry. Variations of patient geometry and tissue density during respiration could reduce the dose accuracy of these 3D optimized plans. To quantify the potential benefits of direct four-dimensional (4D) optimization in robotic lung radiosurgery, 4D optimizations using 1) ray-tracing algorithm with equivalent path-length heterogeneity correction (4EPL(opt)), and 2) Monte Carlo (MC) algorithm (4MC(opt)), were performed in 25 patients. The 4EPL(opt) plans were recalculated using MC algorithm (4MC(recal)) to quantify the dose prediction errors (DPEs). Optimization convergence errors (OCEs) were evaluated by comparing the 4MC(recal) and 4MC(opt) dose results. The results were analyzed by dose-volume histogram indices for selected organs. Statistical equivalence tests were performed to determine the clinical significance of the DPEs and OCEs, compared with a 3% tolerance. Statistical equivalence tests indicated that the DPE and the OCE are significant predominately in GTV D98%. The DPEs in V20 of lung, and D2% of cord, trachea, and esophagus are within 1.2%, while the OCEs are within 10.4% in lung V20 and within 3.5% in trachea D2%. The marked DPE and OCE suggest that 4D MC optimization is important to improve the dosimetric accuracy in robotic-based stereotactic body radiotherapy, despite the longer computation time.