We provide a fully analytical treatment for the partial refrigeration of the thermal motion of a quantum mechanical resonator under the action of feedback. As opposed to standard cavity optomechanics where the aim is to isolate and cool a single mechanical mode, the aim here is to extract the thermal energy from many vibrational modes within a large frequency bandwidth. We consider a standard cold-damping technique, where homodyne readout of the cavity output field is fed into a feedback loop that provides a cooling action directly applied on the mechanical resonator. Analytical and numerical results predict that low final occupancies are achievable independent of the number of modes addressed by the feedback, as long as the cooling rate is smaller than the intermode frequency separation. For resonators exhibiting a few nearly degenerate pairs of modes, cooling is less efficient and a weak dependence on the number of modes is obtained. These scalings hint toward the design of frequency-resolved mechanical resonators, where efficient refrigeration is possible via simultaneous cold-damping feedback.