We fabricate high-resolution diffractive optics on membranes using a bilayer resist system consisting of hydrogensilsesquioxane as a negative electron-beam imaging layer and hardbaked AZPN114 as the underlay. To minimize sidewall etching of the polymer, the AZPN114 layer was etched at -100 degreesC in a cryogenically cooled inductively coupled plasma etcher. Features fabricated on Si supported membrane wafers, where the areas of interest are separated from the platen by the wafer thickness, provide an additional challenge to the low-temperature dry etch process due to low cooling efficiency (and thus membrane heating). Using cooling theory and experimental verification we look at membrane cooling efficiency for different hardware and membrane size combinations. Diffusive cooling in membranes less than 140 mum wide dominates membrane cooling during the etch process. With these small membranes we have fabricated high efficiency x-ray zone plates with linewidths as small as 30 nm and 6:1 aspect ratios.