We introduce a hyperbolic neural network approach to pixel-level active learning for semantic segmentation. Analysis of the data statistics leads to a novel interpretation of the hyperbolic radius as an indicator of data scarcity. In HALO (Hyperbolic Active Learning Optimization), for the first time, we propose the use of epistemic uncertainty as a data acquisition strategy, following the intuition of selecting data points that are the least known. The hyperbolic radius, complemented by the widely-adopted prediction entropy, effectively approximates epistemic uncertainty. We perform extensive experimental analysis based on two established synthetic-to-real benchmarks, i.e. GTAV $\rightarrow$ Cityscapes and SYNTHIA $\rightarrow$ Cityscapes. Additionally, we test HALO on Cityscape $\rightarrow$ ACDC for domain adaptation under adverse weather conditions, and we benchmark both convolutional and attention-based backbones. HALO sets a new state-of-the-art in active learning for semantic segmentation under domain shift and it is the first active learning approach that surpasses the performance of supervised domain adaptation while using only a small portion of labels (i.e., 1%).