Interaction between integral membrane proteins and the lipid-bilayer component of biological membranes is expected to mutually influence the proteins and the membrane. We present quantitative evidence of a manifestation of the lipid-protein interactions in liposomal membranes, reconstituted with actively pumping Na+,K+-ATPase, in terms of nonequilibrium shape fluctuations that contain a relaxation time, tau, which is robust and independent of the specific fluctuation modes of the membrane. In the case of pumping Na+-ions, analysis of the flicker-noise temporal correlation spectrum of the liposomes leads to tau similar or equal to 0.5 s, comparing favorably with an intrinsic reaction-cycle time of about 0.4 s from enzymology.