We develop an experimentally grounded theory for cooling a phonon-coupled microwave cavity far below its ambient bath using a resettable two-qubit reservoir. Motivated by cQED hardware that can operate at the 1–4 K stage, we model a high-$Q$ three-dimensional (3D) cavity repeatedly and weakly engaged by internally correlated two-level pairs that are reprepared and actively reset between interactions. From a Lindblad master equation for a cavity subject to both a phonon bath and a Poisson train of finite-duration ancilla interactions, we obtain closed-form steady states for the cavity photon number and effective temperature in two geometries: (i) only one atom of each pair couples to the cavity, and (ii) both atoms couple collectively. The one-atom geometry cools the cavity below the phonon bath but not below the pair's own temperature. In contrast, when both atoms couple, intrapair coherence renormalizes the upward and downward transition rates seen by the cavity, creating a quantum-enhanced refrigeration effect that can drive the cavity well below the reservoir temperature when phonon damping is weak. Our detuning-aware, time-window–aware collision model (including finite interaction time, arrival rate, and cavity damping) yields analytic lineshapes and identifies broad “cooling valleys” near resonance, together with the crossover between reservoir-dominated and phonon-dominated regimes. Guided by these results, we outline a concrete cQED implementation with two transmons in a 3D cavity using nanosecond flux tuning and fast dissipative or measurement-based resets. Under realistic parameters (MHz-rate cycles, small-angle exchanges, measured reset fidelities), the engineered reservoir reaches $\approx 50$ mK even when the ambient bath is 4 K. Together, these results provide a compact, analyzable route to sub-100-mK microwave modes within a multikelvin cryostat, opening a realistic avenue toward scalable, higher-temperature quantum hardware.

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• Cooling & trapping