Fast emulators of comprehensive climate models are used to explore the impact of anthropogenic emissions in future climate. A new approach to emulators is introduced that predicts distributions of coarse-grained monthly averaged variables as a multivariate Gaussian distribution. The emulator is trained with a state-of-the-art climate model and serves as a good first-order representation for many statistics of future climates. The emulator is applied to statistics of surface temperature and relative humidity for illustrative purposes, but the approach can be applied to any other variable of interest as long the multivariate Gaussian approximation captures the bulk of the distribution. Importantly the emulator accounts for the internal variability of the system, allowing one to examine shifts in distributions of climate variables. In this sense the work can be considered as an extension of pattern scaling emulators that focus on the evolution of the mean rather than the distribution of climate variables.