Some distributed real-time systems interact with external environments that change dynamically, and it is necessary to take the external variables into account when performing task allocation. We developed an approximation algorithm for task allocation, and it finds allocations that are maximally robust against dynamic changes in multiple external variables. Such an algorithm will help to reduce expensive reallocations triggered by changes in unpredictable environments. The algorithm has a polynomial running time, and its robustness optimality is given by an approximation ratio, which equals 2.41 asymptotically, when workloads are large and workload independent utilization of tasks is insignificant.