We develop a computationally efficient scheme to accurately determine finite-temperature band gaps for metal halide perovskites belonging to the class ABX(3) (A = Rb, Cs; B = Ge, Sn, Pb; and X = F, Cl, Br, I). First, an initial estimate of the band gap is provided for the ideal crystalline structure through the use of a range-separated hybrid functional, in which the parameters are determined non-empirically from the electron density and the high-frequency dielectric constant. Next, we consider two kinds of band-gap corrections to account for spin-orbit coupling and thermal vibrations including zero-point motions. In particular, the latter effect is accounted for through the special displacement method, which consists in using a single distorted configuration obtained from the vibrational frequencies and eigenmodes, thereby avoiding lengthy molecular dynamics. The sequential consideration of both corrections systematically improves the band gaps, reaching a mean absolute error of 0.17 eV with respect to experimental values. The computational efficiency of our scheme stems from the fact that only a single calculation at the hybrid-functional level is required and that it is sufficient to evaluate the corrections at the semilocal level of theory. Our scheme is thus convenient for the screening of large databases of metal halide perovskites, including large-size systems.