We previously introduced a preconditioner that has proven effective for hp-FEM dis- cretizations of various challenging elliptic and hyperbolic problems. The construc- tion is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi- Separable matrices. The preconditioner is built as an approximate LDMt factorization through a divide-and-conquer approach. This implies an enhanced flexibility which allows to handle unstructured geometric meshes, anisotropies, and discontinuities. We build on our previous numerical experiments and develop a preconditioner- update strategy that allows us handle time-varying problems. We investigate the performance of the precondition along with the update strategy in context of topology optimization of an acoustic cavity.