Spatial curvature in Unimodular Gravity

We investigate the cosmological implications of unimodular gravity (UG) featuring energy diffusion and spatial curvature. While standard diffusion models often suffer from thermodynamic inconsistencies, we propose a phenomenologically viable power-law Ansatz for the diffusion function, $Q(z) = Q_0(1+z)^\beta$, which strictly satisfies the second law of thermodynamics by demanding positive entropy production ($\beta Q_0 > 0$). Using a joint statistical analysis with the Pantheon+ Type Ia Supernova compilation and Baryon Acoustic Oscillation (BAO) measurements, we tightly constrain the parameter space. We find a diffusion exponent of $\beta = 0.503_{-0.126}^{+0.118}$ and a slight preference for a closed spatial geometry with $\Omega_{k0} = -0.109_{-0.071}^{+0.076}$ at present time. Remarkably, the consideration of spatial curvature and diffusion naturally alleviates the Hubble tension, yielding $H_0 = 73.350_{-0.226}^{+0.221}$ km/s/Mpc while maintaining a consistent cosmic age of $t_0 \simeq 13.61$ Gyr. Furthermore, the constrained diffusion scales as a stable, quintessence-like effective dark energy ($\omega_{\text{eff}} \simeq -0.832$). Thus, unimodular diffusion provides a thermodynamically consistent phenomenological alternative that can alleviate the Hubble tension while preserving both the cosmic age and the sound-horizon scale, with a preference for a closed spatial geometry.