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We present an $O^*(n^3)$ randomized algorithm for estimating the volume of awell-rounded convex body given by a membership oracle, improving on theprevious best complexity of $O^*(n^4)$. The new algorithmic ingredient is anaccelerated cooling schedule where the rate of cooling increases with thetemperature. Previously, the known approach for potentially achieving thisasymptotic complexity relied on a positive resolution of the KLS hyperplaneconjecture, a central open problem in convex geometry.  We also obtain an $O^*(n^3)$ randomized algorithm for integrating a standardGaussian distribution over an arbitrary convex set containing the unit ball.Both the volume and Gaussian volume algorithms use an improved algorithm forsampling a Gaussian distribution restricted to a convex body. In this lattersetting, as we show, the KLS conjecture holds and for a spherical Gaussiandistribution with variance $\sigma^2$, the sampling complexity is$O^*(\max\{n^3, \sigma^2n^2\})$ for the first sample and $O^*(\max\{n^2,\sigma^2n^2\})$ for every subsequent sample.

Gaussian Cooling and O*(n^3) Algorithms for Volume and Gaussian Volume