Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. The numerous approximations for the exchange-correlation energy component in zero-temperature density functional theory, though often used in these high-energy-density simulations with Fermi-weighted electronic densities, are known to miss temperature-dependent effects in

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... he electronic structure of these systems. In this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange-correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb-Oxford inequality, which provides a lower bound for the ground-state exchange-correlation energy, bounds for the exchange-correlation at finite-temperatures are approximated for various degrees of electronic correlation.