This paper studies a version of dark energy built from a generalized entropy called Tsallis entropy and places it inside a modified theory of gravity known as f(R,T) gravity. The authors focus on a holographic dark energy idea, where the amount of dark energy is tied to a horizon size. They use the Hubble horizon as the cutoff and let the dark energy interact with ordinary, pressureless dark matter. Their goal is to see whether this setup can explain the observed late-time acceleration of the universe and how close it behaves to the standard Lambda Cold Dark Matter (ΛCDM) model.

At a high level, Tsallis entropy is a generalization of the usual notion of entropy. When applied to the cosmological horizon, it yields a family of holographic dark energy models (called Tsallis holographic dark energy, or THDE) that differ from the usual Bekenstein-based models. The authors study THDE in two concrete f(R,T) gravity models: a linear model f(R,T)=μR+νT, and a slightly more complex model f(R,T)=R+γR^2+ξT. Here R is the Ricci curvature of spacetime and T is the trace of the matter stress-energy; including T allows a direct coupling between geometry and matter.

The team calculated several standard diagnostic quantities to test the models. They examined the equation of state (EoS) of the dark energy, the deceleration parameter that tracks whether the universe is speeding up or slowing down, and phase-space diagnostics such as the Statefinder pair, the Om(z) diagnostic, the r–q plane, and the wDE–w′DE plane (where w′DE is the derivative of the dark-energy equation of state). These tools help to show whether a model gives late-time acceleration, and how its evolution compares with ΛCDM. The authors report that both f(R,T) models they tested can reach the ΛCDM fixed point in these diagnostics, which means they can mimic the standard model under some conditions.