Topology optimisation of time-dependent problems is inherently time consuming due to having to solve the full forward time history and the full backward adjoint history. Although parallelisation in space has been successfully applied to reduce computational time, traditional time-stepping is inherently serial and limits the possible speed-up. Thus, parallelisation also in the time dimension is required to truly exploit large supercomputers for topology optimisation of time-dependent problems. We apply a space-time finite element method to form an all-at-once system that is solved in parallel using a space-time multigrid preconditioner [1]. Our method has been demonstrated on linear time-dependent heat transfer problems showing up to 52× speed-up compared to traditional time-stepping, although at a 7.7× the cost in terms of core-hours. Modelling of phase-change materials (PCMs) are incorporated using the apparent/effective heat capacity method, while neglecting any natural convection that may arise in the melted PCM. This introduces a highly non-linear heat capacity that spans multiple orders of magnitude. Firstly, this requires modifications to the proposed semi-coarsening scheme [2]. Secondly, a robust non-linear solver is required to solve the full all-at-once system with all non-linearities at the same time. For this, various non-linear solvers are explored, such as Picard iterations, Newton’s method, and Full Approximation Scheme (FAS). Preliminary results, using a simple naïve implementation, indicate that the non-linear problem lowers the obtained speed-up and increases the cost in terms of core-hours. However, these results are expected to improve with better non-linear solvers