Extended Gravity Theories from a Thermodynamic Perspective

We extend the thermodynamic derivation of gravity in the Jacobson framework by generalizing the Clausius relation through a nontrivial entropy functional. We show that entropy deformations appear as modifications of the effective gravitational coupling, defining a broad class of modified gravity theories. However, conventional entropy corrections are insufficient to resolve spacetime singularities within this approach. We then propose a new entropy form by incorporating quantum properties at the level of horizon degrees of freedom. Implementing this entropy in the modified gravitational framework, we study its cosmological implications at both early and late times. In the early Universe, the model predicts a nonsingular phase with a finite Hubble parameter, leading to a de Sitter-like inflationary expansion with finite entropy and temperature. At late times, the theory reproduces, at leading order, the effective dynamics of loop quantum cosmology.