Clustering–the tendency for neighbors of nodes to be connected–quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric to non-geometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finite-size scaling behavior of clustering. Moreover, a slow decay of clustering in the non-geometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.