Emulators of Earth System Models (ESMs) are complementary to ESMs by providing climate information at lower computational costs. Thus far, the emulation of spatially resolved climate extremes has only received limited attention, even though it is one of the most impactful aspects of climate change. Here, we propose a method for the emulation of local annual maximum temperatures, with a focus on reproducing essential statistical properties such as correlations in space and time. We test different emulator configurations and find that driving the emulations with global mean surface temperature offers an optimal compromise of model complexity and performance. We show that the emulations can mimic the temporal evolution and spatial patterns of the underlying climate model simulations and are able to reproduce their natural variability. The general design and the good performance for annual maximum temperatures suggests that the proposed methodology can be applied to other climate extremes.