This paper investigates the thermodynamic evolution of the Universe within the framework of a quadratic equation of state (EOS). Building upon the basis of the quadratic EOS model, as a phenomenological extension to dark energy models, we analyze the implications for cosmic dynamics, including energy density evolution of effective dark matter and dark energy, entropy behavior, and convexity stability conditions. Our approach emphasizes the significance of thermodynamic principles in understanding the late-time acceleration and the crossing of the phantom divide, providing a cohesive description consistent with recent observational data. Moreover, we demonstrate that the $\mathrm{\Lambda }\mathrm{CDM}$ model, regardless of entropy additivity, violates the convexity condition, while the quadratic model aligns with maximum entropy and may prevent a big rip scenario.

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• Cosmology
• Dark energy
• Dark matter
• Evolution of the Universe