Abstract: In this talk, I will discuss the phase of the Euclidean gravity path integral in the presence of a positive cosmological constant. I will begin by reviewing the gravitational partition function on the sphere and clarifying the procedure for determining its precise phase factor. Next, I will discuss how to compute the phase on product manifolds of the form $S^p\times M$, and explain how it connects to physical instabilities in the static patch of $dS_p$. For the special case $p=2$, I will comment on the relation to Jackiw–Teitelboim gravity.