Radiative cooling of carbon cluster anions C(2n+1)(-)C2n+1- (n = 3-5) is investigated using the cryogenic electrostatic ion storage ring DESIREE. Two different strategies are applied to infer infrared emission on slow (milliseconds to seconds) and ultraslow (seconds to minutes) timescales. Initial cooling of the ions over the millisecond timescale is probed indirectly by monitoring the decay in the yield of spontaneous neutralization by thermionic emission. The observed cooling rates are consistent with a statistical model of thermionic electron emission in competition with infrared photon emission due to vibrational de-excitation. Slower cooling over the seconds to minutes timescale associated with infrared emission from low-frequency vibrational modes is probed using time-dependent action spectroscopy. For C(9)(-)C9- and C(11)(-)C11-, cooling is evidenced by the time-evolution of the yield of photo-induced neutralization following resonant excitation of electronic transitions near the detachment threshold. The cross-section for resonant photo-excitation is at least two orders of magnitude greater than for direct photodetachment. In contrast, C(7)(-)C7- lacks electronic transitions near the detachment threshold.