The Hawking temperature in the context of dark energy

An emergent gravity metric incorporating $k-$essence scalar fields $\varphi$ having a Born-Infeld type lagrangian is mapped into a metric whose structure is similar to that of a blackhole of large mass $M$ that has swallowed a global monopole. However, here the field is not that of a monopole but rather that of a $k-$essence scalar field. If ${\varphi }_{emergent}$ be solutions of the emergent gravity equations of motion under cosmological boundary conditions at $\mathrm{\infty }$, then for $r\to \mathrm{\infty }$ the rescaled field $\frac {\phi_{emergent}}{2GM-1}$hasexactcorrespondencewith$\phi$with$\phi(r,t)=\phi_{1}®+\phi_{2}(t)$\mathrm{.}TheHawkingtemperatureofthismetricis$T_{\mathrm emergent}= \frac{\hbar c^{3}}{8\pi GM k_{\mathrm B}}(1-K)^{2}\equiv \frac{\hbar}{8\pi GM k_{\mathrm B}}(1-K)^{2}$,takingthespeedoflight$c=1$. Here $K={\dot{\varphi }}_2^2$ is the kinetic energy of the $k-$essence field $\varphi$ and $K$ is always less than unity, $k_{\mathrm{B}}$ is the Boltzmann constant. This is phenomenologically interesting in the context of Belgiorno {\it et al’s} gravitational analogue experiment.