Understanding the interplay between thermal, elastic, and hydrodynamic effects is crucial for a variety of applications, including the design of soft materials and microfluidic systems. Motivated by these applications, we investigate the emergence of natural convection in a system consisting of a fluid layer that its bottom wall is maintained at temperature $T_h$, and its upper layer is covered by a thin elastic sheet. The sheet is laterally compressed and held at a constant temperature $T_c$, where $T_h > T_c$. Under these conditions, we determine the critical parameters, i.e., the critical temperature difference $T_h - T_c$ and lateral force, at which instability arises in the form of natural convection. We show that for very stiff sheets, the system behaves as if the fluid is confined between two rigid walls, and the solution converges to the classical Rayleigh-B\`enard scenario, where vortices form in the fluid with a typical length scale proportional to the distance between the elastic sheet and the bottom wall. However, for relatively soft sheets, significant deviations from the classical stability curve are observed. Notably, we identify a new branch of solutions characterized by a wavelength that is independent of the fluid layer's thickness but instead depends on the wrinkling wavelength of the elastic sheet. We examine the interplay between this new branch of solutions and the Rayleigh-B\`enard instability, and construct the system's stability state diagram.