Small tungsten cluster anions W-n(-) (n=4-9 and 18-20) are stored in a Penning trap and electronically excited by photoabsorption (E-hv = 1.815, 2.33, 3.5, and 4.66 eV). Delayed electron emission is observed on the milliseconds time scale by systematic variation of the storage duration between laser excitation and ion detection. Even if the photon energy exceeds the electron detachment energy, electrons are emitted several milliseconds after laser excitation. The electron emission time constant is determined as a function of the laser pulse energy. An Arrhenius analysis suggests that the observed delayed electron emission is a thermal process in analogy to thermionic emission of bulk materials. As shown by these experiments there is a simple rule for the dominating cooling channel of laser excited clusters: thermionic emission generally occurs as long as the electron binding energy is lower than the dissociation energy.