Thermodynamic interpretation of the cosmological constant? So I'm going through this paper of emergent gravity where T. Padmanabhan provides a thermodynamic interpretation of Gravity. What I'm failing to understand is what exactly is the thermodynamic interpretation of the cosmological constant in this schema?

This fact also tells us that the cosmological constant problem exists
even at the tree-level of quantum field theory and issues like the
energy of the vacuum, etc. are red herrings. In the alternative
approach, we again use a thermodynamically motivated variational
principle ...

 A: In the emergent gravity program, at least as developed by Padbanabhan, there is no thermodynamical interpretation of the cosmological constant, and the paragraph you quote is trying to say explicitly this.
In short: as the paper goes, you start with a Lagrangian for gravity (this could be the usual Hilbert-Einstein one or certain generalizations). The idea is to separate the Lagrangian in a bulk term and in a boundary term. This boundary term is related to the Noether charge, which the author interprets as entropy (or heat content at some point), and with this interpretation, and the first law of thermodynamics, the paper recovers the usual Einstein field equations. The problem is that this boundary term, which has the physical interpretation of entropy, is invariant under changes of the cosmological constant. Therefore the cosmological constant does not have a correspondent term in usual thermodynamics.
The paragraph you quote is pointing this specifically. The idea being that if the thermodynamical derivation is physical, then the cosmological constant cannot be determined by thermodynamics alone, and the physical value of the constant in our universe does not come from the dynamics of general relativity (this independence from the dynamics is what the author means by saying that the cosmological constant is a problem at tree-level, i.e. classical physics, and that vacuum energy is irrelevant). If you follow the reference at the end of this paragraph [49] it becomes much clearer. In this other paper Padbanabhan argues that the cosmological constant is tied to some fundamental physical scale of the universe. In this thermodynamical scheme for gravity, it would be to say that the Einstein equations are just the first law of thermodynamics but the cosmological constant is like a lattice constant for a crystal, a microscopically feature specific to the system, unlike the first law which is valid for any system large enough
