In this paper, we provide a mathematical description of the onset of instability for buoyancy-driven flow around a cone. The self-similar solutions of the basic flow are derived, where the ambient fluid and the cone have independent temperature gradients. The linear instability properties are investigated by utilizing the Chebyshev collocation method. It is demonstrated that the neutral curves have a two-lobed structure and that changing the Grashof number has only quantitative influence on stability. The critical streamwise location increases when the half-apex angle is increased, and the unstable range of the wave number diminishes. According to the examination of eigenfunction profiles and the progression of the two spatial branches in the (α_{i},α_{r}) planes, the primary instabilities on the surface of cone are identified as type-I mode and type-II mode. The energy analysis is investigated for a typical situation to gain a physical insight, where it is demonstrated that besides the viscous dissipation, the streamline curvature and buoyancy-driven effects are dominant for the type-I mode while inviscid effect plays an essential role in type II. These encouraging results are expected to be conducive to understanding buoyancy-driven systems.