Discrete microstate spectra and Schottky anomaly in charged de Sitter black holes via bumblebee gravity

This paper is devoted to study the thermodynamics of charged de Sitter black holes in Einstein–Bumblebee gravity, a Lorentz-violating extension of general relativity, using an effective two-horizon framework that treats the region between the event and cosmological horizons as a single thermodynamic system. We derive globally consistent thermodynamic variables such as effective temperature, pressure, and volume and show that they satisfy a generalized first law. The effective heat capacity shows a strong Schottky-type anomaly, which is defined by a single-peak structure resulting from the interaction between the two horizons. It is demonstrated here that Lorentz-violating effects cause this feature to undergo systematic and controlled deformations. In particular, we observe that the Lorentz-violation parameter leads to a displacement and rescaling of the heat-capacity peak, which can be interpreted as a change of the effective excitation scale for the thermodynamic response. We also show that the overall thermodynamics of the two-horizon system can be effectively described by a phenomenological finite-level model, where the temperature difference between the horizons acts as an effective excitation gap. Although the map is not derived microscopically, it accurately reproduces the shape of the heat-capacity curve and allows one to interpret the macroscopic thermodynamical enhancement. Our results reveal that Lorentz-violating corrections act like a controlled change to the thermodynamics of black holes with multiple horizons. They keep the overall Schottky-type pattern but change the specific numerical details. This highlights the strong sensitivity of horizon thermodynamics to corrections beyond standard gravity and provides a systematic approach to study these deviations within a semiclassical framework.