We report the experimental observation of a morphological instability of a confined polymer-air double layer sandwiched between two plates set to different temperatures. The homogeneous temperature gradient across the double layer causes the breakup of the polymer film into columns or stripes spanning the two plates. Experimentally, the characteristic wavelength of these patterns varies with the inverse of the initial heat flux through the bilayer. To gain insight into the nature of the instability, we have developed a phenomenological model that describes the heat flow in terms of diffusion through the bilayer. In an idealized microscopic model for the heat flow through the bilayer system, thermal modes in the polymer film with wavelengths ranging from the film thickness to the Debye limit contribute to the heat flux. The low end of this frequency spectrum causes a thermal radiation pressure at the polymer-air surface that destabilizes the film, while the high-frequency modes ascertain the heat conduction through the bilayer.