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In this study, we investigate $f(\bar{R},\bar{T})$ gravity in a {\itnon-standard} theory known as {\bf K-}essence theory, to explore the effect ofthe dark energy in the cosmological scenarios, where $\bar{R}$ is the Ricciscalar and $\bar{T}$ is the trace of the energy-momentum tensor of the {\bfK-}essence geometry. We have used the Dirac-Born-Infeld (DBI) non-standardLagrangian to produce the {\bf K-}essence emergent gravity metric$(\bar{G}_{\mu\nu})$, which is non-conformally equivalent to the gravitationalmetric $(g_{\mu\nu})$. It has been shown that under a flat FLRW backgroundgravitational metric, the modified field equations and Friedmann equations ofthe $f(\bar{R},\bar{T})$ gravity are distinct from the usual ones. To obtainthe equation of state (EoS) parameter $\omega$, we have solved the Friedmannequations by considering $f(\bar{R},\bar{T})\equiv f(\bar{R})+\lambda \bar{T}$,where $\lambda$ is a model parameter. For several forms of $f(\bar{R})$, wehave identified a connection between $\omega$ and time by considering thekinetic energy of the {\bf K-}essence scalar field ($\dot{\phi}^{2}$) as thedark energy density that changes with time. Interestingly, this finding meetsthe requirement of the constraint on $\dot{\phi}^{2}$. We show through graphsof the EoS parameter with time that our model satisfies SNIa+BAO+H(z)observations for a certain time range.

$f(\bar{R},\bar{T})$ gravity in a Non-canonical theory