Open FLRW spacetime in a time-dependent thermodynamic limit

We analyze finite-volume, spatially open (κ=−1) FLRW spacetime(s) in the time-dependent thermodynamic limit, where the physical volume depends on the number of particles N(t) in it and the particle density function γ⁻³(t,T). This prescription makes the boundary a dynamical element and modifies the Friedmann sector through a single combination of the time derivatives of the particle number and the density function. This function accounts for possible thermodynamic and/or quantum changes the system undergoes, and we solve the corresponding FLRW equations written in terms of this function for the proposed finite volume FLRW geometries and different equations of state of the particles. Correspondingly, we present exact trigonometric and hyperbolic solution families on both positive- and negative-energy branches and summarize their kinematic properties. Using an exact reorganization of the Friedmann equation and introducing an effective expansion rate of the volume, a new term similar to a cosmological constant in the equations arises in the framework without adding new fields. The approach also predicts a small, testable correction to observed redshifts sourced by boundary motion. Throughout, the connection between the thermodynamic inputs, which are particle number and density function, and between the background evolution of the open FLRW geometry is kept explicit.