The Gibbs Energy of the Hubble Sphere

We investigate the Gibbs energy of the Hubble sphere in a cosmological model where the observable universe is interpreted as a horizon thermodynamic system, or equivalently as a Carnot-engine-like universe. The enthalpy of the Hubble sphere is taken to be the total horizon mass-energy, $H_H=M_uc^2$, where $M_u=c^3/(GH_0)$ is the Haug-Spavieri universe mass, also interpretable as an extremal universe mass or the extremal Reissner-Nordsr{\"o}m mass. The Hubble sphere is assigned a Hawking-like temperature $T_{Haw,min}=\hbar c/(4\pi k_B R_H)$, where $R_H=c/H_0$, and a horizon entropy $S_H=k_B4\pi R_H^2/l_p^2$. Under these assumptions, the product $T_{Haw,min}S_H$ is exactly equal to the horizon enthalpy, leading to a vanishing Gibbs energy, \begin{equation} G_H=H_H-T_{Haw,min}S_H=0. \end{equation} This result should not be interpreted as implying that the Hubble sphere has zero total energy. Rather, it means that, within the horizon-thermodynamic definitions adopted here, the enthalpy term is exactly balanced by the entropy-temperature term. The Hubble sphere therefore satisfies a zero Gibbs-energy condition. We discuss the relation of this result to extremal Reissner--Nordstrom-like cosmology, zero net horizon entropy, and the Carnot-engine interpretation of the observed cosmic microwave background temperature. We also show that the Gibbs--Helmholtz equation constrains the interpretation of the result: $G_H=0$ should be regarded as a horizon extremality condition or state constraint, not as a globally vanishing Gibbs potential over an unconstrained thermodynamic family.