Lorentz-violating modifications to particle dynamics, thermodynamics and vacuum energy in bumblebee gravity

We investigate how spontaneous Lorentz symmetry breaking in bumblebee gravity modifies particle dynamics, thermodynamics, and vacuum energy around a static black hole background. Starting from the optical-mechanical correspondence, we derive a modified dispersion relation that encodes the influence of the Lorentz-violating parameter $\lambda$ on the propagation of massive and massless modes. We analyze the resulting optical properties, including the effective refractive index, group velocity, and energy-dependent time delay, and show how the non-asymptotically flat geometry reshapes signal propagation. From the same dispersion relation, we construct the interparticle potential for massive and massless excitations and evaluate the electron scattering cross section within the Born approximation, identifying characteristic Lorentz-violating corrections. We then develop a statistical-ensemble description based on the deformed energy-momentum relation and obtain analytic expressions for the thermodynamic observables of a massless bosonic gas. The pressure, mean energy, entropy, and heat capacity are examined in three representative regimes -- extremely close to the horizon, near the photon sphere, and in the asymptotic region -- where Lorentz violation systematically increase the magnitude of these quantities and leads to finite asymptotic plateaus. Finally, we analyze the vacuum state in the curved background and compute the regularized Casimir energy at zero and finite temperature.