Nanoscale oxide resistive switching structures have attracted widespread attention in connection with future logic, memory, neuromorphic computing, and storage structure applications. Resistive switching effects are usually assumed to be caused by conducting filaments (CFs), which are formed by metal diffusion, and their breaking across the insulating oxide between the metal electrodes. Understanding thermal transport in CF containing metal/oxide based structures is critical to determine whether these filaments exist. This paper reports phonon transport in Cu diffused aluminum oxide (AO) in a Cu/AO/Si resistive switching structure where Cu based CFs have formed locally as a result of electric breakdown. This was studied by measuring cross-plane thermal conductivity on CF containing AO layers from 100 to 500 K. Our results suggest that conferring thermal properties upon non-thermoelectric materials as a result of CF formation enables thermal conductivity to be controlled by CF density. Therefore, the proposed methodology, including a local probing method, will also help understand other physical properties of metal/oxide based resistive switching structures in the future.