In multiple target tracking, it is important to be able to evaluate the performance of different tracking algorithms. The trajectory generalized optimal sub-pattern assignment metric (TGOSPA) is a recently proposed metric for such evaluations. The TGOSPA metric is computed as the solution to an optimization problem, but for large tracking scenarios, solving this problem becomes computationally demanding. In this letter, we present an approximation algorithm for evaluating the TGOSPA metric, based on casting the TGOSPA problem as an unbalanced multimarginal optimal transport problem. Following recent advances in computational optimal transport, we introduce an entropy regularization and derive an iterative scheme for solving the Lagrangian dual of the regularized problem. Numerical results suggest that our proposed algorithm is more computationally efficient than the alternative of computing the exact metric using a linear programming solver, while still providing an adequate approximation of the metric.