Heavy-heavy-light-light (HHLL) correlators of pairwise identical scalars in CFTs with a large central charge in any number of dimensions admit a double scaling limit where the ratio of the heavy conformal dimension to the central charge becomes large as the separation between the light operators becomes null. In this limit the stress tensor sector of a generic HHLL correlator receives contributions from the multi stress tensor operators with any number of stress tensors, as long as their twist is not increased by index contractions. We show how one can compute this leading twist stress tensor sector when the conformal dimension of the light operators is large and the stress tensor sector approximates the thermal CFT correlator. In this regime the value of the correlator is related to the length of the spacelike geodesic which approaches the boundary of the dual asymptotically AdS spacetime at the points of light operator insertions. We provide a detailed description of the infinite volume limit. In two spacetime dimensions the HHLL Virasoro vacuum block is reproduced, while in four spacetime dimensions the result is written in terms of elliptic integrals.
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