Equipartition energy, Noether energy and boundary term in gravitational action

In the study of horizon thermodynamics and emergent gravity two natural expressions for energy, E = 2T S (equipartition energy) and E = T S (Noether energy) arise which differ by a factor 2. I clarify the role of these two expressions in different contexts and show how E = T S is also closely related to the Noether charge arising from the boundary term of the Einstein-Hilbert action.