We report on the measurement of the 7Be(n,p)7Li cross section from thermal to approximately 325 keV neutron energy, performed in the high-flux experimental area (EAR2) of the n_TOF facility at CERN. This reaction plays a key role in the lithium yield of the big bang nucleosynthesis (BBN) for standard cosmology. The only two previous time-of-flight measurements performed on this reaction did not cover the energy window of interest for BBN, and they showed a large discrepancy between each other. The measurement was performed with a Si telescope and a high-purity sample produced by implantation of a 7Be ion beam at the ISOLDE facility at CERN. While a significantly higher cross section is found at low energy, relative to current evaluations, in the region of BBN interest, the present results are consistent with the values inferred from the time-reversal 7Li(p,n)7Be reaction, thus yielding only a relatively minor improvement on the so-called cosmological lithium problem. The relevance of these results on the near-threshold neutron production in the p+7Li reaction is also discussed.