<p>Recent experiments have implemented resetting by means of a time-varying external harmonic trap, whereby the trap stiffness is changed in finite-time and the system is reset when it relaxes to an equilibrium distribution in the final trap. Such setups are very similar to those studied in the context of the finite-time Landauer erasure principle. In this work, we analyze the thermodynamic costs of such a setup by deriving a moment generating function for the work cost of recurrently changing the trap stiffness, thereby maintaining a non-equilibrium steady state. For this heretofore unstudied case, we obtain explicit expressions for the mean and variance of the work both for a specific experimentally viable protocol as well as an optimal protocol which minimizes the mean cost. For both these procedures, our analysis captures both the large-time and short-time corrections. For the optimal protocol, we obtain a closed form expression for the mean cost for all protocol durations, thereby making contact with earlier work on geometric measures of dissipation-minimizing optimal protocols that implement information erasure.</p>