  In this research, we introduce and investigate an approximation method that
preserves the structural integrity of the non-isothermal
Cahn-Hilliard-Navier-Stokes system. Our approach extends a previously proposed
technique [1], which utilizes conforming (inf-sup stable) finite elements in
space, coupled with implicit time discretization employing convex-concave
splitting. Expanding upon this method, we incorporate the unstable P1|P1 pair
for the Navier-Stokes contributions, integrating Brezzi-Pitk\"aranta
stabilization. Additionally, we improve the enforcement of incompressibility
conditions through grad div stabilization. While these techniques are
well-established for Navier-Stokes equations, it becomes apparent that for
non-isothermal models, they introduce additional coupling terms to the equation
governing internal energy. To ensure the conservation of total energy and
maintain entropy production, these stabilization terms are appropriately
integrated into the internal energy equation.
