In this paper, we propose a local projection stabilization (LPS) finite element method applied to numerically solve natural convection problems. This method replaces the projection-stabilized structure of standard LPS methods by an interpolation-stabilized structure, which only acts on the high frequencies components of the flow. This approach gives rise to a method which may be cast in the variational multi-scale framework, and constitutes a low-cost, accurate solver (of optimal error order) for incompressible flows, despite being only weakly consistent. Numerical simulations and results for the buoyancy-driven airflow in a square cavity with differentially heated side walls at high Rayleigh numbers (up to Ra=107) are given and compared with benchmark solutions. Good accuracy is obtained with relatively coarse grids.