Gravity and on-shell probe actions

In any gravitational theory and in a wide class of background space-times, we argue that there exists a simple, yet profound, relation between the on-shell Euclidean gravitational action and the on-shell Euclidean action of probes. The probes can be, for instance, charged particles or branes. The relation is tightly related to the thermodynamic nature of gravity. We provide precise checks of the relation in several examples, which include both asymptotically flat and asymptotically AdS space-times, with particle, D-brane and M-brane probes. Perfect consistency is found in all cases, including in a highly non-trivial example including α^′-corrections.