  We prove results relating the theory of optimal transport and generalized
Ricci flow. We define an adapted cost functional for measures using a solution
of the associated dilaton flow. This determines a formal notion of geodesics in
the space of measures, and we show geodesic convexity of an associated entropy
functional. Finally, we show monotonicity of the cost along the backwards heat
flow, and use this to give a new proof of the monotonicity of the energy
functional along generalized Ricci flow.
