Scalar$-$Tensor Gravity as a Probe of Generalized Black Hole Entropy

We develop a geometric realization of a broad class of generalized black hole entropy functionals by establishing their direct correspondence with the Misner$-$Sharp quasilocal mass and the Wald Noether$-$charge entropy in scalar$-$tensor theories of gravity. The resulting models feature a scale-dependent effective gravitational coupling, whose functional dependence is determined by the underlying entropy parameters. Within this framework, we derive explicit Einstein-frame scalar potentials: for Barrow entropy, a steep exponential potential; for Tsallis$-$Cirto entropy, an exponential potential governed by the nonextensivity parameter; and for quantum-gravity and entanglement$-$induced corrections, an approximately linear potential. These distinct potentials generate characteristic cosmological phenomenology, with implications for inflationary dynamics, late-time dark-energy behavior, and non-singular bouncing cosmologies. The framework is compatible with current constraints from solar-system tests, big-bang nucleosynthesis, and pulsar-timing observations, and it yields predictions that can be probed by forthcoming observational surveys. In this way, the analysis establishes a unified and geometrically grounded connection between information$-$theoretic entropy proposals and gravitational field theory.