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Due to the conformal factor problem, the definition of the Euclideangravitational path integral requires a non-trivial choice of contour. Thepresent work examines a generalization of a recently proposed rule-of-thumb\cite{Marolf:2022ntb} for selecting this contour at quadratic order about asaddle. The original proposal depended on the choice of an indefinite-signaturemetric on the space of perturbations, which was taken to be a DeWitt metricwith parameter $\alpha =-1$. This choice was made to match previous results,but was otherwise admittedly {\it ad hoc}. To begin to investigate the physicsassociated with the choice of such a metric, we now explore contours definedusing analogous prescriptions for $\alpha \neq -1$. We study such contours forEuclidean gravity linearized about AdS-Schwarzschild black holes in reflectingcavities with thermal (canonical ensemble) boundary conditions, and we comparepath-integral stability of the associated saddles with thermodynamic stabilityof the classical spacetimes. While the contour generally depends on the choiceof DeWitt parameter $\alpha$, the precise agreement between these two notionsof stability found at $\alpha =-1$ continues to hold over the finite interval$(-2,-2/d)$, where $d$ is the dimension of the bulk spacetime. This agreementmanifestly fails for $\alpha > -2/d$ when the DeWitt metric becomes positivedefinite. However, we also find dramatic failures for $\alpha< -2$ thatcorrelate with breakdowns of the de Donder-like gauge condition defined by$\alpha$, and at which the relevant fluctuation operator fails to bediagonalizable. This provides criteria that may be useful in predicting metricson the space of perturbations that give physically-useful contours in moregeneral settings. Along the way, we also identify an interesting error in\cite{Marolf:2022ntb}, though we show this error to be harmless.

Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter