In this work, we introduce an integrated depth measure for functional data defined over complex multidimensional domains. We consider functional data whose discrete realizations are irregularly spaced, and may be available only over portions of the domain. To address this issue, we propose an integrated depth based on a Voronoi tessellation of the multidimensional domain. This approach ensures favorable statistical properties for the proposed depth, as well as computational efficiency, enabling the analysis of large-scale functional datasets. We validate our proposal with the study of air temperatures across the Earth surface, as provided by the CESM2 Large Ensemble Community Project. The proposed depth is able to capture the increase in global temperatures since the 1980s, coherently with global warming.