We study the swelling and shrinking behavior of core-shell microgels adsorbed on silicon wafers. In these systems, the core is made of cross-linked poly(N-isopropylmethacrylamide) and the shell consists of cross-linked poly(N-n-propylacrylamide). In suspension, these particles exhibit an extended linear swelling behavior in the temperature interval between the lower critical solution temperatures of the two polymers. Using ellipsometry and atomic force microscopy, we show that this linear response is also observed in the adsorbed state.