Physical Viability of Little Rip Cosmology in $$\boldsymbol{f(R,\Sigma,T)}$$ Gravity within the FLRW Framework

We study the evolution of the universe using several Rip cosmologies within the framework of $$f(R,\Sigma,T)$$ gravity, utilizing spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) models. There are three types of Rip models. Employing the Little Rip (LR) scenario, we derive exact expressions for the pressure, energy density, and the equation of state parameter and investigate their time evolution. Particular emphasis is placed on examining the physical acceptability of the model through the behavior of key quantities such as the energy conditions (NEC, DEC, and SEC) and the transition of pressure from positive to negative values, which naturally explains the shift from decelerated to accelerated expansion. The analysis demonstrates that the energy density remains positive and decreases with cosmic time, while the pressure evolves from a radiation-like regime to a dark-energy-dominated phase. Moreover, violation of the strong energy condition, a necessary feature for late-time cosmic acceleration, is explicitly realized. Our findings indicate that the the LR scenario in $$f(R,\Sigma,T)$$ gravity provides a viable description of the late-time universe, avoids finite-time singularities, and offers an alternative explanation to standard dark energy models.