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This study aims to investigate the impact of dark energy in cosmological scenarios by exploiting $f(\bar{R},\bar{T})$ gravity within the framework of a {\it non-standard} theory, called {\it {\bf K-}essence} theory, where $\bar{R}$ represents the Ricci scalar and $\bar{T}$ denotes the trace of the energy-momentum tensor associated with the {\bf K-}essence geometry. The Dirac-Born-Infeld (DBI) non-standard Lagrangian has been employed to generate the emergent gravity metric $(\bar{G}_{μν})$ associated with the {\bf K-}essence. This metric is distinct from the usual gravitational metric $(g_{μν})$. It has been shown that under a flat FLRW background gravitational metric, the modified field equations and the Friedmann equations of the $f(\bar{R},\bar{T})$ gravity are distinct from the usual ones. In order to get the equation of state (EOS) parameter $ω$, we have solved the Friedmann equations by taking into account the function $f(\bar{R},\bar{T})\equiv f(\bar{R})+λ\bar{T}$, where $λ$ represents a parameter within the model. We have found a relationship between $ω$ and time for different kinds of $f(\bar{R})$ by treating the kinetic energy of the {\bf K-}essence scalar field ($\dotϕ^{2}$) as the dark energy density which fluctuates with time. Surprisingly, this result meets the condition of the restriction on $\dotϕ^{2}$. By presenting graphical representations of the EOS parameter with time, we show that our model is consistent with the data of $SNIa$+$BAO$+$H(z)$ within a certain temporal interval.