  We consider a thermodynamically consistent model for the evolution of
thermally conducting two-phase incompressible fluids. Complementing previous
results, we prove additional regularity properties of solutions in the case
when the evolution takes place in the two-dimensional flat torus with periodic
boundary conditions. Thanks to improved regularity, we can also prove
uniqueness and characterize the long-time behavior of trajectories showing
existence of the global attractor in a suitable phase-space.
