Einstein-Born-Infeld-AdS Black Holes: Quantum Thermodynamics, Gravitational Lensing, and Heat Engine Analysis

We investigate the thermodynamic and optical properties of Einstein-Born-Infeld-Anti-de Sitter (EBI-AdS) black holes (BHs). Our study derives the Hawking temperature using the standard surface gravity method and examines quantum corrections through both the Generalized Uncertainty Principle (GUP) and exponential entropy modifications, revealing enhanced thermal radiation and possible remnant formation scenarios. The gravitational redshift analysis disentangles the contributions of mass, cosmological constant, electromagnetic charge, and Born-Infeld (BI) nonlinear corrections, with the latter scaling as a 4 /r 6 and therefore becoming significant only in the near-horizon regime. Using the Gauss-Bonnet theorem, we compute the weak deflection angle of light in both vacuum and plasma media and show that dispersive effects may either enhance or suppress nonlinear electrodynamic signatures depending on the observational setup. In the extended phase-space formalism, where the BH mass is interpreted as enthalpy, the thermodynamic analysis reveals phase structures characterized by heat-capacity transitions between positive and negative regions, indicating local stability and instability depending on the parameter space. We further analyze BH heat engines operating in rectangular cycles and find efficiencies in the range η 0.11–0.21, corresponding to approximately 30–61% of the associated Carnot efficiencies, consistent with other AdS BH systems. A comparison with Johnson's framework shows that BI-induced corrections to the heat-engine efficiency are typically of order 10 -12 for standard parameter choices, although they become appreciable in the strong-field regime where r h ≲ 1.5 in Planck units. Finally, the plasma-lensing analysis reveals frequency-dependent refractive modifications encoded in the plasma parameter, thereby offering an additional observational channel for testing nonlinear electromagnetic effects in black-hole environments.