We consider properties of the gravitational path integral, \({\mathcal{Z}}_{\text{grav}}\) , of a four-dimensional gravitational effective field theory with Λ > 0 at the quantum level. To leading order, \({\mathcal{Z}}_{\text{grav}}\) is dominated by a four-sphere saddle subject to small fluctuations. Beyond this, \({\mathcal{Z}}_{\text{grav}}\) receives contributions from additional geometries that may include Einstein metrics of positive curvature. We discuss how a general positive curvature Einstein metric contributes to \({\mathcal{Z}}_{\text{grav}}\) at one-loop level. Along the way, we discuss Einstein-Maxwell theory with Λ > 0, and identify an interesting class of closed non-Einstein gravitational instantons. We provide a detailed study for the specific case of \({\mathbb{C}}{P}^{2}\) which is distinguished as the saddle with second largest volume and positive definite tensor eigenspectrum. We present exact one-loop results for scalar particles, Maxwell theory, and Einstein gravity about the Fubini-Study metric on \({\mathbb{C}}{P}^{2}\) .
