Gauss–Bonnet effects in $$f(R,\Sigma ,T)$$ gravity

Abstract We investigate the cosmological dynamics of Parameterized Absolute Parallelism (PAP) geometry within the framework of modified $$f(R,\Sigma ,T)$$ f ( R , Σ , T ) gravity. By generalizing the Gauss–Bonnet invariant $${\mathcal {G}}_b$$ G b using the PAP curvature tensor $$B^{\alpha }{}_{\mu \nu \sigma },$$ B α μ ν σ , we incorporate both Riemannian curvature and torsion effects into a unified framework. The modified field equations are derived for a spatially flat FLRW background, and exact analytical solutions for the Hubble parameter, scale factor, energy density, and pressure are obtained under a constant equation-of-state parameter $$\omega .$$ ω . Our analysis shows that the PAP deformation parameter b and matter–torsion coupling $$\eta $$ η significantly modify cosmic expansion, the effective equation of state, and curvature diagnostics. Furthermore, we explore the role of $${\mathcal {G}}_b$$ G b in torsion-driven Gauss–Bonnet inflation and demonstrate how PAP corrections alter the slow-roll dynamics, enabling sustained inflation even for flat potentials. A detailed stability analysis reveals the critical regions in the $$(b,\eta )$$ ( b , η ) parameter space separating stable and unstable cosmological phases. This unified approach recovers General Relativity, teleparallel gravity, and minimally coupled PAP models as limiting cases, offering new insights into early- and late-time acceleration driven by torsion and higher-curvature effects.