It depends. The Intergovernmental Panel on Climate Change's (IPCC) Assessment Report Six (AR6) took an ambitious step towards ending so-called 'model democracy' by discounting climate models that are too warm over the historical period (i.e., models that "run hot") when making projections of global temperature change. However, the IPCC did not address whether this procedure is reliable for other physical quantities. Here we demonstrate the dangers of weighting climate models according to their skill in reproducing historical global-mean surface temperature using three other climate variables of interest: annual average precipitation change, regional average temperature change, and regional average precipitation change. We find that the IPCC's weighting scheme leads to an improved prediction of global average precipitation. However, a different story emerges on regional scales, where we find a heterogeneous pattern of error reduction in future regional precipitation. This stands in sharp contrast with the broad regional pattern of error reduction in future temperature projections, though we do find some regions where error is not significantly reduced. Our results demonstrate that practitioners using weighted climate model ensembles for climate projections must take care, lest they produce biased climate projections that result from an inappropriate weighting procedure.