The dependence of the spectral width of the longitudinal Localized Surface Plasmon Resonance (LSPR) of individual gold nanorods protected by a silica shell is investigated as a function of their size. Experiments were performed using the Spatial Modulation Spectroscopy technique which permits to optically determine both the spectral characteristics of the LSPR of an individual nanoparticle and its morphology. The measured LSPR is shown to broaden with reduction of both the length and diameter of the nanorod, in contrast with the predictions of existing classical and quantum theoretical models. This behavior can be reproduced assuming the LSPR width linearly depends on the inverse of an effective length proportional to the square root of the particle surface, with the same slope as that recently determined for silica-coated silver nanospheres.