In this work, we explore the applications of geometric flows and the Raychaudhuri equation in gravitational theory. This equation, which governs the evolution of the expansion scalar, describes how the volume spanned by a bundle of nearby geodesics evolves in curved space-time. By introducing the concept of geometric flow, we can reinterpret the physical meaning of the expansion scalar and identify an interesting connection between an entropy-like quantity and the mean geodesic deviation. This notion of geometric entropy could be particularly relevant in the study of the geometry of black holes coupled to nonlinear matter fields; an example is provided.