We construct a geometric structure within the space of Hyperboloid models: a Homogeneous Flat F-manifold. Flat F-manifolds, characterized by a commutative, associative multiplication on the tangent bundle with a compatible flat connection, provide a versatile framework for studying integrable systems and algebraic structures in geometry. In contrast, statistical manifolds, which combine Riemannian metrics with dual affine connections, form the foundation of information geometry and the study of probabilistic spaces. Motivated by the rich interplay between these structures, we establish a connection between the Homogeneous Flat F-manifold and the standard statistical manifold structure in the Hyperboloid space through a generalized Legendre transform. This connection highlights potential applications in both theoretical and applied geometry.