I'm familiar with the Lagrangian for linearized gravity about a flat background,
$$ \mathcal{L} = \frac{1}{2}[(\partial_\mu h^{\mu\nu} \partial_\nu h - \partial_\mu h^{\rho \sigma} \partial_\rho h^\mu_\sigma + \frac{1}{2} (\partial_\mu h^{\rho\sigma})^2 - \frac{1}{2} (\partial_\mu h)^2] + \mathcal{O}(h^3) $$
See, for example, Carroll's textbook, $\S 7.1$.
I am interested in the Lagrangian for linearized gravity around a curved background. My background metric is diagonal and the only source is a cosmological constant. In principle I know this can be done by simply expanding the Einstein-Hilbert action to order $h^2$ about the background, but this is tedious and I would hope there is a reference where this has already been worked out.
This review by Flanagan and Hughes gives the Einstein and Ricci tensors to linear order in $h$ ($\S 5$) but not the action. Additionally, this comment on a question about linearized gravity mentions that in a curved background the action has a term of the form $R_{\mu\nu\rho\sigma} h^{\mu\nu} h^{\rho\sigma}$, where $R_{\mu\nu\rho\sigma}$ is the curvature of the background metric, but does not elaborate further.
Is there a reference that clearly states the action for linearized gravity about a curved background?