In project scheduling applications, data is often affected by various sources of uncertainty. Robust optimization allowing to find reliable planned schedules may be decisive to overcome imperfect data knowledge. The notion of anchor robustness, as introduced in the literature for PERT, makes it possible to guarantee schedule dates for a subset of jobs under processing time uncertainty. In this paper, anchor robustness is extended to a time-dependent variant of PERT where job completion times are functions of their starting times. It is applied to a generalization of processing time uncertainty as well as to uncertain non-availability periods. Three problems involving anchorage are considered in the case of budgeted uncertainty: checking that a given subset of jobs is anchored in a given schedule, finding a schedule of minimal makespan in which a given subset of jobs is anchored and finding an anchored subset of maximum weight in a given schedule. Polynomial time algorithms are proposed for the first two problems while inapproximability results are given for the third one.