Quantum-gravitational effects on Joule-Thomson expansion and black hole shadows in F(R) gravity with barrow entropy corrections

We present a novel investigation into the thermodynamic properties and shadow images of a topological phantom AdS black hole within the framework of $F\left(R\right)$ gravity, utilizing the Barrow entropy formulation. We introduce a detailed study of the Joule-Thomson expansion, calculating the Joule-Thomson coefficient and mapping isenthalpic and inversion curves to characterize heating/cooling phases. Our findings reveal that the Barrow parameter plays a crucial role in altering the shape and position of these thermodynamic curves, thereby significantly impacting the black hole's thermal behavior. Moreover, we demonstrate that variations in $F\left(R\right)$ gravity parameters such as the scalar curvature $R_0$, the type of field interaction η, and $f_{R_0}$—lead to substantial modifications in the shadow of the black hole. These changes in shadow size and shape underscore the direct influence of the underlying gravitational theory on the interaction between black holes and light, offering new perspectives on how black holes are perceived by distant observers. Additionally, we calculate the total observed intensities from the emission function of the accretion disk, with graphical analysis showing that the $F\left(R\right)$ gravity parameters markedly affect the intensity distribution around the black hole. This has significant implications for the accretion disk's emission properties and the resulting shadow image, highlighting the critical impact of modified gravity theories on observable black hole phenomena. This work offers fresh insights and underscores the importance of considering modified gravity frameworks when studying black hole thermodynamics and optical properties.