Thermodynamics and geometrical optics of Reissner–Nordström–de Sitter black holes in noncommutative geometry

We investigate the thermodynamic, optical, and dynamical properties of Reissner–Nordström–de Sitter black holes in a noncommutative spacetime endowed with a fundamental minimal length scale Θ Θ . Working within a consistent two-horizon framework, we formulate an effective first law of thermodynamics and introduce an entanglement entropy that captures correlation effects between the event and cosmological horizons. Imposing the lukewarm condition, under which both horizons share a common temperature, uniquely fixes the entropy correction and yields closed-form expressions for the temperature, pressure, and electric potential. An analysis of the heat capacity, entropy, Helmholtz free energy, and pressure reveals a noncommutativity-induced second-order phase transition, highlighting the crucial role of short-distance structure in the critical behavior of the system. On the optical side, we examine photon motion and weak gravitational lensing in this geometry. We show that noncommutativity deforms the photon effective potential and shifts the critical impact parameter separating capture from scattering. Using the Gauss–Bonnet method, we derive an analytical expression for the weak deflection angle and demonstrate how charge and the minimal length scale modify the optical response in the presence of a cosmological constant. Finally, we establish a direct link between geometry and dynamics by analyzing the Lyapunov exponent of circular photon orbits and the imaginary part of the quasinormal mode frequencies. We show that the black hole parameters leave distinct and systematic imprints on both orbital instability and damping: increasing the mass suppresses the instability and leads to longer-lived modes, charge enhances both the divergence of null geodesics and the decay rate, the cosmological constant slightly reduces damping, and noncommutativity amplifies the instability and accelerates relaxation.