Road travel costs are important knowledge hidden in large-scale GPS trajectory data sets, the discovery of which can benefit many applications such as intelligent route planning and automatic driving navigation. While there are previous studies which tackled this task by modeling it as a regression problem with spatial smoothness taken into account, they unreasonably assumed that the latent cost of each road remains unchanged over time. Other works on route planning and recommendation that have considered temporal factors simply assumed that the temporal dynamics be known in advance as a parametric function over time, which is not faithful to reality. To overcome these limitations, in this paper, we propose an extension to a previous static trajectory regression framework by learning the temporal dynamics of road travel costs in an innovative non-parametric manner which can effectively overcome the temporal sparsity problem. In particular, we unify multiple different trajectory regression problems in a multi-task framework by introducing a novel cross-task regularization which encourages temporal smoothness on the change of road travel costs. We then propose an efficient block coordinate descent method to solve the resulting problem by exploiting its separable structures and prove its convergence to global optimum. Experiments conducted on both synthetic and real data sets demonstrate the effectiveness of our method and its improved accuracy on travel time prediction.