Unsteady-state heat transfer can be observed in many engineering applications, where it is inherently time-dependent and cannot be fully characterised by steady-state analysis. Incorporating transient effects into topology optimisation remains computationally demanding due to the need for fine temporal resolution and the complex interaction between spatial and temporal discretisation errors [1]. Space-time finite element methods offer an alternative by treating space and time in a unified variational framework, enabling improved numerical robustness and flexibility for transient problems [2]. This study presents an adaptive space-time finite element formulation integrated with density-based topology optimisation for transient heat conduction. The governing equations are formulated in a fully space-time weak form, allowing unstructured discretisation and local refinement in both spatial and temporal dimensions. A goal-oriented adaptive strategy is employed to refine the space-time mesh in regions and time intervals that most strongly influence the optimisation objective, thereby reducing computational cost while maintaining accuracy. Design sensitivities are consistently derived within the space-time framework and coupled with PDE-based filtering to ensure well-posed optimisation problems. 2+1D numerical examples demonstrate that the proposed method efficiently captures complex transient thermal behaviour and yields optimised designs that differ from those obtained using conventional steady-state analysis.