Modified entropy from action principle

We propose a modified gravity theory by extending the Einstein-Hilbert action with an arbitrary function of the Ricci scalar and the Kretschmann scalar invariants. The resulting modified Friedmann equations for a spatially flat FRW universe are derived, which remain free of higher-order derivatives and reduce to the standard Friedmann equations in the limiting case. Employing the gravity-thermodynamics conjecture, we investigate the thermodynamic behavior at the apparent horizon and derive the corresponding modified entropy. Using the first law of thermodynamics together with the modified Friedmann equations, we obtain a general expression for the apparent horizon entropy. This formalism allows us to compute the modified entropy for various well-known entropy models. Our approach establishes a consistent thermodynamic framework linking modified gravity theories constructed from curvature invariants to generalized entropy functions on the cosmological apparent horizon.