Unimodular random graphs with Property (T) have cost one

Hutchcroft and Pete showed that countably infinite groups with Property (T) admit cost one actions, resolving a question of Gaboriau. We give a streamlined proof of their theorem, and extend it both to locally compact second countable groups and unimodular random graphs. We prove unimodular random graph analogues of the Connes–Weiss and Glasner–Weiss theorem characterising Property (T).