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Anti-Stokes photoluminescence (ASPL), which is an up-conversion phonon-assisted process of the radiative recombination of photoexcited charge carriers, was investigated in methylammonium lead bromide (MALB) perovskite nanocrystals (NCs) with mean sizes that varied from about 6 to 120 nm. The structure properties of the MALB NCs were investigated by means of the scanning and transmission electron microscopy, X-ray diffraction and Raman spectroscopy. ASPL spectra of MALB NCs were measured under near-resonant laser excitation with a photon energy of 2.33 eV and they were compared with the results of the photoluminescence (PL) measurements under non-resonant excitation at 3.06 eV to reveal a contribution of phonon-assisted processes in ASPL. MALB NCs with a mean size of about 6 nm were found to demonstrate the most efficient ASPL, which is explained by an enhanced contribution of the phonon absorption process during the photoexcitation of small NCs. The obtained results can be useful for the application of nanocrystalline organometal perovskites in optoelectronic and all-optical solid-state cooling devices.