In this contribution, we apply adaptive isogeometric analysis to a phase‐field model for topology optimization. To increase the efficiency of the computation, we perform local mesh refinement and coarsening between the time increments of the simulation. To provide a trial solution for the iterative solver of the next increment, state variables have to be transferred from the old to the new mesh. We therefore compare the application of a discrete least squares fit with an L 2 ‐projection in well known two‐ and three‐dimensional benchmarks and analyze the influence of these methods on the quality of the results.