The famous IPCC general circulation models (GCMs) constitute the primary tools for climate projections. Calibrating, or tuning the parameters of the models can significantly improve their predictions, thus their scientific and societal impacts. Unfortunately, traditional tuning techniques remain time-consuming and computationally costly, even at coarse resolution. A specific challenge for the tuning of climate models lies in the tuning of both fast and slow climatic features: while atmospheric processes adjust on hourly to weekly timescales, vegetation or ocean dynamics drive mechanisms of variability at decadal to millennial timescales. In this work, we explore whether and how History Matching, which uses machine learning based emulators to accelerate and automate the tuning process, is relevant for tuning climate models with multiple timescales. To facilitate this exploration, we work with a climate model of intermediate complexity, yet test experimental tuning protocols that can be directly applied to more complex GCMs to reduce uncertainty in climate projections.