# Multiple saddles in path integral

I am reading the Jerusalem Lectures by Harlow. On page 44 he calculates the thermal partition function using the path integral with no matter fields, $$Z(\beta) = \int \mathcal{D}[g] e^{-I_E[g]}.$$ The calculation is done using saddle-point techniques $$Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}]},$$ where $$g_\text{cl}$$ are classical solutions of the Einstein equations. I am confused by this sum as I have not seen a similar case in which one expands around multiple saddles in ordinary QFT.

I have read several articles where the approximation is taken around a single saddle $$g_\text{cl}$$. This is achieved by writing $$g = g_\text{cl} + \delta g$$ and expanding the action as $$I_E[g] = I_E[g_\text{cl}] + I_{E,2}[\delta g] + \dots$$ to obtain $$\log Z = -I_E[g_\text{cl}] + \log\int \mathcal{D}[\delta g] e^{-I_{E,2}[\delta g]} + \dots$$ If we include only the first term, we get $$Z(\beta) \approx e^{-I_E[g_{cl}]},$$ which agrees with the sum formula for one saddle. How can we show that the sum formula holds for multiple saddles?

1. Ref. 1 uses the method of steepest descent with multiple stationary/saddle points/instanton sectors in the semiclassical limit $$\hbar\to 0$$, cf. e.g. this related Phys.SE post.