High-order finite element (FE) discretizations are increasingly used for large-scale simulations. They deliver high accuracy per degree of freedom and arithmetic intensity on GPU-accelerated architectures. Their effectiveness, however, is tightly coupled to mesh quality and element distribution, which become challenging for curved, high-order meshes and for optimization problems with evolving, localized features. We adopt a PDE-constrained R-refinement (R-adaptivity) formulation in which high-order mesh nodal coordinates are treated as control variables and are optimized on a fixed-topology mesh. The objective is represented as a convex combination of a Target-Matrix Optimization Paradigm (TMOP) mesh-quality metric and a physics-driven measure of FE solution accuracy, subject to the discrete governing PDE. Consistent gradients with respect to mesh motion are obtained by adjoint sensitivity analysis, while a convolution-type filter regularizes the displacement field to mitigate localized sensitivities and avoid element tangling. The resulting large-scale optimization is solved using gradient-based methods, combined with a Jacobian-positivity line search to ensure mesh validity. For representative Poisson and linear elasticity problems, R-refinement concentrates resolution in regions most critical to accuracy and yields up to an order-of-magnitude reduction in discretization error at fixed element count. Building on this foundation, we integrate R-refinement into density-based topology optimization implemented with the SiMPL methodology. Instead of relying solely on H-refinement, the mesh is continuously re-positioned during the density update while preserving mesh connectivity and parallel decomposition. This coupling enables robust high-order discretizations of compliance and constraint functionals, improves interface resolution in the density field, and maintains well-shaped curvilinear elements suitable for high-order analysis. The implementation leverages MFEM’s high-order finite element infrastructure and GPU-friendly operator evaluation, and we demonstrate scalable high-order topology optimization runs on HPC GPU machines.