Dual matter induced oscillatory dynamics and thermodynamic analysis around black holes

In this work, we investigate the dynamical and thermodynamic characteristics of a static, spherically symmetric black hole (BH) immersed in an anisotropic fluid within the framework of an extended theory of gravity. The spacetime geometry is specified by the metric function h(r), which depends on the parameters a, b, k, and w, as well as on the cosmological constant Λ. In this case, using the Hamiltonian formalism, the motion of neutral test particles is analyzed through the effective potential, leading to analytical expressions for the specific energy, angular momentum, and effective force. The position of the innermost stable circular orbits (ISCOs) is obtained, showing that increasing a, b, and w deepens and narrows the potential well, moving the ISCOs closer to the horizon and strengthening the gravitational attraction compared to the Schwarzschild case. Small perturbations around stable orbits yield the fundamental frequencies, which increase with the coupling parameters, suggesting a link with quasi-periodic oscillations (QPOs). In this context, using the Barrow entropy formalism, we compute the geometric mass, Hawking temperature, and entropy, showing that a, b, k, w, and Λ affect the thermal stability and emission rate of the BH.