The Thermodynamic Origin of Time in Unified Quantum Gravity

Here, we demonstrate that cosmic time emerges from holographic entropy in Unified Quantum Gravity (UQG). Through numerical analysis of the Friedmann equations and Bekenstein-Hawking entropy, we establish that physical time t is mathematically equivalent to the square root of cosmic horizon entropy SH​ in the matter-dominated era: t∝SH​ ​ with correlation coefficient R2=0.9993. In the dark energy-dominated era, this relationship breaks down as entropy saturates, with the time-entropy exponent γ increasing from 0.5 to 0.772. Asymptotic analysis reveals that γ→1 in the far future, indicating convergence to a thermodynamic equilibrium state (De Sitter phase) rather than thermal death. This result operationalizes the profound claim that time emerges from entropy,'' providing a concrete mathematical framework where time is not a fundamental coordinate but a counter of quantum states. The transition from $\gamma = 0.5$ to $\gamma = 1$ represents a cosmic phase transition from computational efficiency (matter era) to thermodynamic stability (dark energy era), with the universe evolving toward a thermodynamic time crystal''---a stable state where time flows but entropy remains constant. Our analysis bridges thermodynamics and general relativity, offering a new perspective on the arrow of time and the ultimate fate of the universe.