The k-essence emergent Reissner-Nordstrom-de Sitter spacetime has exactly mapped on to the Robinson-Trautman (RT) type spacetime with cosmological constant Λ for certain configuration of k-essence scalar field. Theoretically, we evaluated that the thermodynamical quantities for the RT type emergent black hole is different from the usual one in the presence of kinetic energy of the k-essence scalar field i.e., the dark energy density. We restrict ourselves into the fact that the dark energy density (K) is to be unity, then the effective temperature and pressure both are negative for the RT type emergent black hole which implies that the system is thermodynamically unstable when the charge Q = 0 and the emergent spacetime is only dark energy dominated and it does not radiate when Q = 0. The thermodynamically unstable situation is physically plausible only when we consider spin degrees of freedom of a system. We have made this analysis in the context of dark energy in an emergent gravity scenario having k-essence scalar fields φ with a Dirac-Born-Infeld type lagrangian. The scalar field also satisfies the emergent equation of motion at r → ∞.

For a particular type of k-essence scalar field, the k-essence emergent gravity metric is exactly mapped on to the Barriola-Vilenkin (BV) type metric for Schwarzschild background established by Gangopadhyay and Manna. Based on the S. Chandrasekhar, we report the exciting features of the time-like geodesic structure in the presence of dark energy in an emergent gravity scenario for this Barriola-Vilenkin type metric. We trace the different kinds of trajectories for time-like geodesic in the presence of dark energy for the k-essence emergent Barriola-Vilenkin spacetime, which are same with the Schwarzchild spacetime in view of the basic orientation, but the allowed ranges of the aphelion and perihelion distances are much more different. The bound and unbound orbits are plotted for a fixed value of the dark energy density. II. REVIEW OF K-ESSENCE THEORY AND EMERGENT GRAVITYIn this section, we present a short review of the kessence theory and construction of the effective emergent metric. The k-essence scalar field φ minimally coupled to the background spacetime metric g µν has action [9]- [13] S k [φ, g µν ] = d 4 x √ −gL(X, φ)

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