Feedback cooling enables a system to achieve low temperatures through measurement-based control. Determining the thermodynamic cost required to achieve the ideal cooling efficiency within a finite time remains an important problem. In this work, we establish a thermodynamic uncertainty relation (TUR) for feedback cooling in classical underdamped Langevin systems, thereby deriving a trade-off between the cooling efficiency and the entropy reduction rate. The obtained TUR implies that simultaneous achievement of the ideal cooling efficiency and finite entropy reduction rate is asymptotically possible by letting the fluctuation of the reversible local mean velocity diverge. This is shown to be feasible by using a feedback control based on the Kalman filter. Our results clarify the thermodynamic costs of achieving the fundamental cooling limit of feedback control from the perspective of the TUR.

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• Information thermodynamics
• Stochastic thermodynamics