Thermodynamics of BTZ Black Holes in Bumblebee Gravity with Barrow Entropy with Cavity-Modification

We investigate the thermodynamics of $(2+1)$ dimensional BTZ black holes in the Einstein bumblebee gravity theory with spontaneous breaking of Lorentz symmetry caused by a nonzero vacuum expectation value of the vector field. We analyse corrections to black hole thermodynamics, including the non extensive Barrow entropy, parametrising quantum gravitational corrections to the Bekenstein Hawking area law. We introduce the York cavity formalism by placing the black hole in a finite isothermal cavity to obtain a properly defined canonical ensemble. Here we arrive at the corrected temperature, free energy, and stability conditions of the BTZ black hole, demonstrating the interplay among the Lorentz violating effects, the Barrow corrections to Entropy, and the boundary conditions within the cavity. Our results indicate that the collective impact of bumblebee dynamics and Barrow entropy significantly alters the phase structure, equilibrium configurations, and thermal stability of BTZ black holes. These findings provide insight into the implications of Lorentz violation and generalised entropy frameworks in lower dimensional quantum gravity.