Thermodynamic Cosmology from the CMB Temperature and the Planck Temperature and the Relation to Quantum Cosmology

We develop a thermodynamic formulation of the specific RHt=ct cosmological framework presented here in which the CMB temperature, the Planck temperature, and a horizon Hawking-temperature scale provide direct expressions for a broad set of cosmological quantities. Our preferred geometrical realization identifies the Hubble boundary with the event horizon of the neutral Haug–Spavieri (HS) minimal solution, whose lapse function is algebraically equivalent to that of an extremal Reissner–Nordström horizon but admits a charge-free interpretation. With RHt=GMt/c2 and the conjectured growth law dMt/dt=mp/tp=c3/G, the horizon evolves as RHt=ct. Although the conventional net Hawking flux of an extremal horizon vanishes, the Carnot-universe interpretation retains counteracting outward and inward thermodynamic channels whose signed fluxes cancel externally; their Hawking-temperature magnitude remains relevant to the geometric-mean relation Tcmb,t=TmaxTmin,t. In the HS branch, this relation gives the standard temperature scaling when the proposed redshift mapping 1+z=Ht/H0 is adopted; establishing that mapping observationally requires its derivation from photon propagation in the completed dynamical metric. We also formulate a conservative FLRW/apparent-horizon alternative, in which RA,t=c/Ht=2ct during a radiation-type phase and the same temperature scaling follows from 1+z=a0/a. In natural units, the background relations reduce to a compact one-scale description through the reduced Compton wavelength of the horizon mass. The framework provides a direct and testable connection between horizon thermodynamics, the CMB temperature, and the principal cosmological scales.