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In applications of climate information, coarse-resolution climate projections commonly need to be downscaled to a finer grid. One challenge of this requirement is the modeling of sub-grid variability and the spatial and temporal dependence at the finer scale. Here, a post-processing procedure is proposed for temperature projections that addresses this challenge. The procedure employs statistical bias correction and stochastic downscaling in two steps. In a first step, errors that are related to spatial and temporal features of the first two moments of the temperature distribution at model scale are identified and corrected. Secondly, residual space-time dependence at the finer scale is analyzed using a statistical model, from which realizations are generated and then combined with appropriate climate change signal to form the downscaled projection fields. Using a high-resolution observational gridded data product, the proposed approach is applied in a case study where projections of two regional climate models from the EURO-CORDEX ensemble are bias-corrected and downscaled to a 1x1 km grid in the Trondelag area of Norway. A cross-validation study shows that the proposed procedure generates results that better reflect the marginal distributional properties of the data product and have better consistency in space and time than empirical quantile mapping.