We report differential I-band and J-band photometry of S Ori 45, a cool (spectral type M8.5), young (1-8 Myr) brown dwarf of the sigma Orionis cluster with a likely mass estimated at around 20 times the mass of Jupiter. We detect variability (amplitudes ranging from 34 to 81 mmag) and observe a modulation at a period of 2.5-3.6 h in both optical and near-infrared light curves. The most recent optical data set, however, presents a modulation at the very short period of 46.4 +/- 1.5 min, which remains a mystery. The origin of the 2.5-3.6 h modulation is analyzed in terms of various scenarios: inhomogeneous features (dust clouds or magnetically induced dark spots) co-rotating with the object's surface, and presence of an unseen very low-mass companion that is steadily transferring mass to the primary. Because of the very young age of the object and its persistent strong Hα emission, the possible presence of an accreting disk is also discussed. If the period of a few hours is related to rotation, our results suggest that sigma Orionis low-mass brown dwarfs are rotating faster than more massive cluster brown dwarfs at a rate consistent with their theoretically inferred masses and radii, implying that all of these objects have undergone similar angular momentum evolution. Based on observations made with the Isaac Newton Telescope (INT) operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias; with the 2.2 m telescope at the German-Spanish Astronomical Center at Calar Alto in Spain; with the Nordic Optical Telescope of the Observatorio del Roque de Los Muchachos; and with the 1.5 m Carlos Sánchez Telescope operated on the island of Tenerife in the Spanish Observatorio del Teide.