The Hawking temperature in the context of dark energy

An emergent gravity metric incorporating $k-$essence scalar fields $ϕ$ 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 $ϕ_{emergent}$ be solutions of the emergent gravity equations of motion under cosmological boundary conditions at $\infty$, then for $r\rightarrow\infty$ the rescaled field $\frac {ϕ_{emergent}}{2GM-1}$ has exact correspondence with $ϕ$ with $ϕ(r,t)=ϕ_{1}(r)+ϕ_{2}(t)$. The Hawking temperature of this metric is $T_{\mathrm emergent}= \frac{\hbar c^{3}}{8πGM k_{\mathrm B}}(1-K)^{2}\equiv \frac{\hbar}{8πGM k_{\mathrm B}}(1-K)^{2}$, taking the speed of light $c=1$. Here $K=\dotϕ_{2}^{2}$ is the kinetic energy of the $k-$essence field $ϕ$ 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.

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