This paper presents an implementation of phantom-cell adaptive mesh refinement (AMR) on a graphics processing unit (GPU) using CLAMR, a cell-based mini-app that runs on a variety of next-generation platforms. Phantom-cell AMR is a hybrid method of cell-based AMR and patch-based AMR that provides a separation of physics and mesh codes. By designing a structure that allows each level of the mesh to be independent, there are minimal development requirements that are needed to convert regular grid applications to AMR. The decoupling of physics and mesh codes through these phantom cells improves composability and creates an easy pathway toward implementing AMR codes on Exascale systems, specifically targeting GPUs. Physics and mesh codes can be accelerated individually, allowing for fewer dependencies and more opportunities for optimization. A complete implementation of phantom-cell AMR on a GPU with opencl is presented for the purpose of showing the simplicity of porting the algorithms to accelerator-based architectures and the performance and optimization improvements that are made as a result.