  In this study, we explore a spherically symmetric charged black hole with a
cosmological constant under the influence of a Kalb--Ramond field background.
We compute the photon sphere and shadow radii, validating our findings using
observational data from the Event Horizon Telescope (EHT), with a particular
emphasis on the shadow images of Sagittarius $A^{*}$. Furthermore, we
investigate the \textit{greybody} factors, emission rate, and partial
absorption cross section. It is shown that the Lorentz-violating parameter
\(\bar{l}\) has an important effect on the absorption cross section. Our
analysis also includes an examination of the topological charge,
temperature-dependent topology, and generalized free energy. In particular, we
regard the AdS charged black hole with an antisymmetric tensor background as a
topological defect in the thermodynamic space, then the system has the same
topological classification to the charged RN-AdS black hole.
