Recently, a novel class of modified gravity theories has been proposed, wherein Einstein’s General Relativity (GR) is extended by incorporating a quadratic energy–momentum term of the form $T_{\mu \nu }T^{\mu \nu } $ , coupled via a constant parameter $\alpha $ . The corresponding field equations deviate from the Einstein equations only in the presence of matter. Analytical studies indicate that, without interaction, the energy-momentum squared term remains subdominant, mainly enabling non-singular Big Bang scenarios. In this work, we investigate this framework in a homogeneous and isotropic cosmological background. We show that, in its minimal form, the theory does not naturally explain late-time cosmic acceleration. Although a cosmological constant can remedy this, it introduces an effective dark energy component with positive pressure during the matter era, distorting large-scale structure formation. To overcome this, we derive an analytical dark energy form by redefining its equation of state and imposing boundary conditions consistent with early- and late-time cosmology. The resulting phenomenological model alleviates the coincidence and fine-tuning problems and ensures classical stability. Observational constraints confirm good agreement with current data, though a statefinder analysis shows that, while the model mimics $\Lambda $ CDM today, it deviates in the far future as the acceleration rate increases.
