We use density functional theory with periodic boundary conditions to investigate diffusion of Al, O, Pt, Hf, and Y on the α-Al₂O₃(0001) surface as a simple model to obtain insight into possible diffusion mechanisms occurring at alumina grain boundaries (GBs) in thermal barrier coatings (TBCs). We calculate diffusion pathways, activation energies, and diffusion constants. For the elements involved in alumina growth, we find that Al diffusion is facile and involves a simple hop from one most stable adsorption site to the next without local minima in between whereas O diffusion has a much higher barrier and has local minima along the minimum energy pathway. This trend is consistent with relative rates of diffusion at alumina GBs. Regarding common dopants in the metal (bond coat) alloy substrate from which alumina grows, we predict that Pt diffusion is facile but thermodynamically disfavored relative to Pt remaining in the bond coat alloy, while Hf and Y diffusion involve much higher barriers than Al diffusion. We use these results to rationalize some observations regarding the role of reactive elements Hf and Y as well as Pt in TBCs.