Goal-oriented adaptive algorithms produce optimal grids to solve challenging engineering problems. Recently, a novel error representation using (unconventional) pseudo-dual problems for goal-oriented adaptivity in the context of frequency-domain wave-propagation problems has been developed. In this paper, we extend this error representation to the case of time-domain problems. We express the entire problem in weak form in order to derive the adjoint formulation and apply goal-oriented adaptivity. One dimensional (1D) numerical results show that upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies.