Constraining Lorentz Violation in Kalb-Ramond Gravity via Thermodynamics and Gravitational Wave Analysis

We investigate the observational signatures of a static, spherically symmetric black hole embedded in a spontaneous Kalb-Ramond (KR) background. By normalizing the solution to the physically observable mass $M_{\text{phys}}$, we demonstrate that the thermodynamics of the KR black hole are consistent with General Relativity, with no deviations in the entropy-area law. However, the Lorentz-violating parameter $l$ induces distinct geometric effects: it suppresses the optical shadow radius by a factor of $\sqrt{1-l}$ and hardens the quasinormal mode frequency by the inverse factor. Utilizing Event Horizon Telescope (EHT) data for Sagittarius A*, and assuming the mass prior derived from stellar dynamics, we place a constraint of $l \lesssim 0.19$. While the product of the shadow radius and ringdown frequency remains degenerate with General Relativity, the specific suppression of the shadow size offers a viable pathway to constrain Planck-scale physics with current and future horizon-scale imaging.