Variable-density incompressible Navier-Stokes equations are used in several fluid dynamics problems, particularly for modelling the interaction between two non-miscible fluids with different densities. They constitute a mixed PDE system that couples two kinds of equations: on one hand, the mass conservation equation which guarantees the mass density of each fluid to remain unchanged during the motion, and, on the other hand, the set of incompressible Navier-Stokes equations which describe the motion of the fluids. The present work aims to give a hybrid formulation for this problem: the mass conservation equation is discretized through a finite volume scheme coupled with a low-order Hybrid High-Order (HHO) method for the Navier-Stokes equations. The main advantage of such a formulation is that it mimics the continuous problem's physical properties, namely energy conservation. Moreover, the computational cost can be drastically reduced by eliminating degrees of freedom inside the cells through static condensation.