  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 $(\bar{G}_{\mu\nu})$ associated with the {\bf
K-}essence. This metric is distinct from the usual gravitational metric
$(g_{\mu\nu})$. 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 $\omega$, we have solved the
Friedmann equations by taking into account the function
$f(\bar{R},\bar{T})\equiv f(\bar{R})+\lambda \bar{T}$, where $\lambda$
represents a parameter within the model. We have found a relationship between
$\omega$ and time for different kinds of $f(\bar{R})$ by treating the kinetic
energy of the {\bf K-}essence scalar field ($\dot{\phi}^{2}$) as the dark
energy density which fluctuates with time. Surprisingly, this result meets the
condition of the restriction on $\dot{\phi}^{2}$. 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.
