The flat FLRW metric in static spherical Schwarzschild/Droste-style coordinates is
$$g_{\mu \nu}=\left(
\begin{array}{cccc}
\rm c^2-H^2 r^2 & 0 & 0 & 0 \\
0 & \frac{1}{\rm H^2 r^2/c^2-1} & 0 & 0 \\
0 & 0 & \rm -r^2 & 0 \\
0 & 0 & 0 & \rm -r^2 \sin ^2 \theta \\
\end{array}
\right)$$
or in Gullstrand/Painlevé-style proper distance raindrop coordinates
$$g_{\mu \nu}=\left(
\begin{array}{cccc}
\rm c^2-H^2 r^2 & \rm H \ r & 0 & 0 \\
\rm H \ r & -1 & 0 & 0 \\
0 & 0 & \rm -r^2 & 0 \\
0 & 0 & 0 & \rm -r^2 \sin ^2 \theta \\
\end{array}
\right)$$
or the cartesian form with $\rm \{ t,x,y,z\}$ instead of $\rm \{ t, r, \theta, \phi \}$
$$g_{\mu \nu}=\left(
\begin{array}{cccc}
\rm c^2-H^2 (x^2+y^2+z^2) & \rm H \ x & \rm H \ y & \rm H \ z \\
\rm H \ x & -1 & 0 & 0 \\
\rm H \ y & 0 & \rm -1 & 0 \\
\rm H \ z & 0 & 0 & \rm -1 \\
\end{array}
\right)$$
but if $\rm H$ is not constant and you have a time dependend Hubbleparameter the whole metric becomes time dependend, so that is only static when the other densities are neglible compared to the dark energy density. Then we have basically De Sitter space where $\rm H=c \sqrt{\Lambda/3}$ is constant, or Minkowski if you also set $\Lambda=0$.
In our universe this will be the case in a few billion years when the Hubble radius becomes asymptotically constant, see here at around $\rm t > 35 \ Gyr$ where $\rm H$ doesn't really change anymore. During inflation the Hubbleradius $\rm c/H$ is also expected to be constant, so that era can also be described in static coordinates.
For $\rm H$ to change you need $\rho$ to change, and then your curvature
$$ \rm K=12 (2 H^4+2 H^2 \dot{H}+\dot{H}^2)/c^4$$
is no longer constant (since the $\rm \dot{H}=dH/dt\neq 0$) and the metric is no longer static, but rather $\rm t$-dependend. With $\rm \dot{H}=0$ the curvature invariant reduces to $\rm K=24H^4/c^4$ and the requirement of constant curvature that your reference presumes for static coordinates is fulfilled. I haven't read it though, but since you quoted it yourself:
Octaaf quoted Florides: "the only FLRW spacetimes that are expressible in static form are the 6 FLRW spacetimes of constant curvature"
I don't expect it to be different there.
