Define a physical free energy for evolving FLRW horizons at constant Kodama-Hayward temperature

Problem statement

Work in four-dimensional Einstein gravity with $c=\hbar=k_B=1$ and fixed $G$. Consider the expanding flat FLRW family $$H(a)=H_\star+B a^{-2},\qquad \dot H=-2H(H-H_\star),\qquad H_\star>0,\ B>0,$$ with dimensionless $a$. Its apparent horizon has $r_A=H^{-1}$, $S_{\rm BH}=\pi/(GH^2)$ and a constant geometrical temperature $T_{\rm KH}=H_\star/(2\pi)$, although its area changes. This is the constant-temperature family studied in [1,2]. For the realization with $f_\Lambda=0$, define $\rho=3H^2/(8\pi G)$ and $p_e=-\rho-\dot H/(4\pi G)$, so that the effective stress tensor is conserved. With $V=4\pi/(3H^3)$, $E_{\rm bulk}=\rho V$ and $W_e=(\rho-p_e)/2$, the background equations imply $$-\dot E_{\rm bulk}+W_e\dot V=T_{\rm KH}\dot S_{\rm BH}.$$ Postulating $E_H=2T_{\rm KH}S_{\rm BH}$ and defining $F_H=E_H-T_{\rm KH}S_{\rm BH}$ then gives $$F_{\rm geo}=T_{\rm KH}S_{\rm BH}=\frac{H_\star}{2GH^2},\qquad -\dot E_{\rm bulk}+W_e\dot V=\dot F_{\rm geo}.$$ This algebraic identity is established. The unresolved problem is whether $F_{\rm geo}$ is the free energy of a physically specified gravitational system, or which independently derived potential and additional flux or work terms replace it. Construct a boundary-based thermodynamic description of this evolving family that determines its internal energy, entropy and free energy independently of the identity above. Use spherical timelike boundaries $R_\eta(t)=\eta/H(t)$ with $0<\eta<1$, and state which side of each boundary belongs to the system. The admissible framework is Einstein gravity with a covariantly conserved matter realization of the displayed effective stress tensor, the gravitational boundary and corner terms needed for a well-posed variational problem, and a fixed reference prescription. Boundary observers, their time normalization, matter boundary conditions and any reservoir must be specified. Reference terms must be fixed local functionals of the declared boundary data, not functions fitted along one cosmological history. The target is the physical meaning and domain of validity of the horizon limit, not an arbitrary choice of zero energy. A successful result should explain whether the relevant ensemble or controlled nonequilibrium description exists, whether its temperature is related to $T_{\rm KH}$ by a derived normalization, and whether its free-energy balance reduces to the displayed relation. If it does not, determine the correction or establish an obstruction for the stated class of boundary formulations. The distinction matters because a geometrical Smarr relation does not by itself establish a statistical ensemble [8]. Moreover, the ordinary pure-de-Sitter Euclidean saddle has dimensionless action $I_E=-S_{\rm dS}$ and canonical energy zero [3-5], whereas the positive potential $F_{\rm geo}$ uses a different horizon-energy prescription. These are not automatically contradictory observables, but an explicit map between their systems and boundary terms is required. A resolution would determine when constant horizon temperature supports an actual thermodynamic free energy during cosmological evolution and when the language describes only a useful background identity.

Current state

The literature checked for this proposal through 2026-09-07 establishes several parts of the problem, but does not provide the complete construction specified below. Komatsu [2] derived the constant-$T_{\rm KH}$ background. The linked paper [1] derives the density/volume decomposition and the conditional free-energy relation, while explicitly treating horizon equipartition as an assumption. Its Appendix C proposes additional action/entropy assumptions rather than specifying a boundary ensemble. The numerical curves and the algebraic balance therefore do not resolve the present question. Banihashemi and Jacobson [3] define cosmological-horizon canonical and microcanonical ensembles using a York boundary and discuss stability, including an evolving reservoir. Banihashemi et al. [4] explain the static-de-Sitter first-law sign using the distinction between Brown-York internal energy and matter Killing energy. These are essential benchmarks. The open target here is the exact nonstationary flat-FLRW family with an evolving apparent horizon, including a controlled relation between its process balance and state variations; it is not the already-studied static sign puzzle. Arias, Diaz and Sundell [5] give the standard negative Euclidean action in the pure-de-Sitter limit and distinguish an observer construction with boundary contributions. Pavon [6] studies horizon equipartition for de Sitter and power-law universes, obtains formal free energies, and explicitly assumes a canonical interpretation in the de Sitter example. Thus positive horizon-energy conventions must be compared with a specified ensemble, rather than dismissed solely from the sign of a different energy. Wu et al. [8] derive dynamical Smarr relations and a chemical-work interpretation. Those results show that dynamical energy and work relations already exist; a new rearrangement of a Smarr formula alone would not resolve this challenge. Zhao [7] develops corrected entropy and first laws to leading order around stationary cosmological event horizons. The present target is an apparent horizon over a finite evolving interval, not only a leading perturbation of a stationary event horizon. What remains is a single specified formulation that supplies independent thermodynamic potentials, their state variables, the boundary-to-horizon limit and all process fluxes for this exact family, or a theorem delimiting its failure. The distinction between variation among equilibrium states and differentiation along a dissipative solution is central. No claim is made that every possible gravitational ensemble or every matter completion has been classified.

Resolution criteria

A resolving paper must satisfy all of the following. 1. Specify the system and independent construction. Give a covariant matter realization of the stated background stress tensor, the boundary action or equivalent Hamiltonian formulation on $R_\eta=\eta/H$, observers, time normalization, reference terms, thermodynamic controls, and reservoir assumptions. Obtain the relevant energy and entropy from that formulation. Postulating $E_H=2TS$, setting $F=TS$, or integrating the target process identity to define $F$ is not an independent construction. 2. Establish a state interpretation or its precise failure. For a canonical construction, derive $Z$, $F=-T\log Z$, $E=-\partial_\beta\log Z$ and $S=\beta(E-F)$ with $\beta=1/T$ and all other controls held fixed. Check stability using the correct fixed controls; an ordinary positive canonical measure must obey $\partial_\beta^2\log Z={\rm Var}(E)\ge0$. For another ensemble or a nonequilibrium formulation, state the replacement relations, justify them and explain the limit in which a Helmholtz potential exists. A geometrical temperature alone is insufficient. 3. Derive and evaluate the process balance. Keep all boundary work, matter flux and reservoir terms, distinguish state variations from time derivatives, and derive the limit $\eta\to1^-$ with its observer normalization. Determine whether the resulting physical potential reproduces $F_{\rm geo}=H_\star/(2GH^2)$ and the linked paper's free-energy relation. Otherwise calculate the discrepancy from the independent formulation and identify its cause. Give an analytic result or a controlled approximation with an explicit error bound covering the finite regime $5/4\le H/H_\star\le2$, as well as the limit $H/H_\star\to1$. A result valid only at exact de Sitter is insufficient. 4. Match the known limits without silently changing the system. Recover the chosen York-boundary de Sitter thermodynamics and explain its relation to $I_E=-S_{\rm dS}$ and canonical $E=0$ when the standard pure-de-Sitter system is recovered. Identify every boundary or matter contribution responsible for a different energy. Keep the apparent and event horizons distinct. Demonstrate which conclusions are invariant under the explicitly allowed reference prescriptions and which remain convention dependent. 5. Supply a checkable outcome. Either give the construction and its domain of validity, with a derived corrected balance if necessary, or prove an obstruction for the stated class of boundary formulations and exhibit the nonintegrable term, inconsistent ensemble condition or unavoidable physical flux responsible. Failure of one arbitrary energy convention, a single numerical trajectory, the static action sign alone, or absence of a global Killing vector alone does not settle the challenge. Provide the derivation and reproducible symbolic or numerical checks needed to audit the result.

Citations

1. N. Komatsu, Holographic thermodynamic relation for dissipative and non-dissipative universes in a flat FLRW cosmology, arXiv:2408.11279v2 (2024), especially Sections III-IV and Appendix C.
2. N. Komatsu, Evolution of thermodynamic quantities on cosmological horizon in Lambda(t) model, Physical Review D 108, 083515 (2023), arXiv:2306.11285.
3. B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons, Journal of High Energy Physics 07 (2022) 042, arXiv:2204.05324.
4. B. Banihashemi, T. Jacobson, A. Svesko and M. Visser, The minus sign in the first law of de Sitter horizons, Journal of High Energy Physics 01 (2023) 054, arXiv:2208.11706.
5. C. Arias, F. Diaz and P. Sundell, De Sitter Space and Entanglement, Classical and Quantum Gravity 37, 015009 (2020), arXiv:1901.04554v4, especially Section 5.
6. D. Pavon, Equipartition of Energy in Gravitating Systems, arXiv:2504.09500 (2025).
7. J. Zhao, The entropy of dynamical de Sitter horizons, European Physical Journal C 85, 750 (2025), arXiv:2503.16138.
8. S.-F. Wu, B. Wang, X.-H. Ge and G.-H. Yang, Gravitational thermodynamics and universal holographic duality in dynamical spacetimes, arXiv:1109.0193.

Linked papers

• Holographic thermodynamic relation for dissipative and non-dissipative universes in a flat FLRW cosmologyrelated7 Sept 2026
• Boundary energy and detector nonstationarity at constant-temperature FLRW horizonsdeclared at submission7 Sept 2026