Diffractive Fresnel zone plates made of tungsten show great promise for focusing hard x-ray free-electron laser (XFEL) radiation to very small spot sizes. However, they have to withstand the high-intensity pulses of the beam without being damaged. This might be problematic since each XFEL pulse will create a significant temperature increase in the zone plate nanostructures and it is therefore crucial that the optics are thermally stable, even for a large number of pulses. Here we have studied the thermal stability of tungsten zone-platelike nanostructures on diamond substrates using a pulsed Nd:YAG laser which creates temperature profiles similar to those expected from XFEL pulses. We found that the structures remained intact up to a laser fluence of 100 mJ cm(-2), corresponding to a 6 keV x-ray fluence of 590 mJ cm-2, which is above typical fluence levels in an unfocused XFEL beam. We have also performed an initial damage experiment at the LCLS hard XFEL facility at SLAC National Accelerator Laboratory, where a tungsten zone plate on a diamond substrate was exposed to 105 pulses of 6 keV x-rays with a pulse fluence of 350 mJ cm-2 without any damage occurring.