We present thermopower measurements on a high electron mobility two-dimensional electron gas (2DEG) in a thin suspended membrane. We show that the small dimension of the membrane substantially reduces the thermal conductivity compared with bulk material so that it is possible to establish a strong thermal gradient along the 2DEG even over a distance of a few micrometers. We find that the zero-field thermopower is significantly affected by the micropatterning. In contrast to 2DEGs incorporated in a bulk material, the diffusion contribution to the thermopower remains dominant up to temperature of 7 K, until the phonon drag becomes strong and governs the thermopower. We also find that the coupling between electrons and phonons in the phonon-drag regime is due to screened deformation potentials, in contrast to the piezoelectric coupling found for bulk phonons.