= Solution
Let $K=\partial_t$ and let the future unit normal to $\Sigma_\tau$ be $n=f^{-1}\partial_t$. The conserved Killing energy is the flux
$$
E(\tau)=\int_{\Sigma_\tau}T_{ab}n^aK^b\sqrt h\,d^3x.
$$
Substitution of the <Klein-Gordon scalar stress-energy tensor> gives exactly
$$
E(\tau)=\frac12\int_{\Sigma_\tau}
\left[f^{-1}(\partial_t\psi)^2
+fh^{ij}\partial_i\psi\partial_j\psi+f\mu^2\psi^2\right]
\sqrt h\,d^3x.
$$
Apply the <divergence theorem> to the slab $[\tau_1,\tau_2]\times\mathbb R^3$. The spatial-boundary flux vanishes because $\psi$ decays, and $\nabla_aJ^a=0$ makes the two time-slice fluxes equal. Thus this <conserved scalar-field energy in a static spacetime> is independent of $\tau$.
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