Gauge-fixing condition invariant under auxiliary gauge transformation

Question

In quantum gravity one usually splits the metric $g= \bar{g}+h$ into a background field $\bar{g}$ and a fluctuation field $h$. In order to obtain a propagator one has to gauge fix the action (e.g. Einstein-Hilbert) which breaks the gauge symmetry $$h\rightarrow h+\mathcal{L}g$$ $$\bar{g}\rightarrow \bar{g}$$ where $\mathcal{L}$ is the Lie derivative.
However, one can choose now the gauge fixing such that it is invariant under $$h\rightarrow h+\mathcal{L}h$$ $$\bar{g}\rightarrow \bar{g} +\mathcal{L}\bar{g}.$$ My problem is now that I don’t see why for example the de-Donder gauge condition $$ F_\mu[\bar{g},h]=\bar{\nabla}^\nu h_{\mu\nu}-\frac{1}{2}\bar{\nabla}_\mu h $$ is invariant under the latter transformation.
References:

1. J. Pawlowski, M. Reichert Quantum gravity: A fluctuating point of view (eq. 46)

2. N. Christiansen From the Quark-Gluon Plasma to Quantum Gravity (p. 35)

Answer 1

1. The (infinitesimal) background gauge transformation is $$\begin{align} \delta_B\bar{g}_{\mu\nu} ~=~& {\cal L}_X\bar{g}_{\mu\nu},\cr \delta_Bh_{\mu\nu} ~=~& {\cal L}_Xh_{\mu\nu}, \end{align} \tag{46} $$ where $X$ is an (infinitesimal) vector field and ${\cal L}_X$ denotes the Lie derivative.

2. More generally, $$ \delta_Bf\left(\bar{g}_{\mu\nu},\partial_{\lambda}\bar{g}_{\mu\nu}, \ldots; h_{\mu\nu},\partial_{\lambda}h_{\mu\nu}, \ldots\right)~=~{\cal L}_Xf\left(\bar{g}_{\mu\nu},\partial_{\lambda}\bar{g}_{\mu\nu}, \ldots; h_{\mu\nu},\partial_{\lambda}h_{\mu\nu}, \ldots\right),$$ at least for differential geometric objects.

3. The de-Donder gauge covector $F_{\mu}$ in eq. (7) is not invariant but transforms as $$ \delta_B F_{\mu}~=~{\cal L}_X F_{\mu}~=~X[F_{\mu}]+\partial_{\mu}X^{\nu}F_{\nu}.$$

4. The gauge-fixing Lagrangian density term $$ \rho~=~-\frac{1}{2\xi}\sqrt{|\bar{g}|} F_{\mu} \bar{g}^{\mu\nu}F_{\nu} \tag{6}$$ is not strictly invariant but transforms by a total spacetime derivative $$\delta_B\rho~=~{\cal L}_X\rho~=~\rho{\rm div}_{\rho} X~=~d_{\mu}(\rho X^{\mu}). $$

5. One of the main features of the background field method is that the background gauge transformation $\delta_B$ is a gauge quasi-symmetry of the full gauge-fixed action.

References:

1. J.M. Pawlowski & M. Reichert, Quantum gravity: a fluctuating point of view, arXiv:2007.10353.