Every regular convex cone $K \subset \mathbb{R}^{n+1}$ has a canonical barrier $F$ and a canonical Hessian metric $g=D^2F$. We study the Riemannian Langevin equation on $(K,g)$, establishing several relations between Brownian motion on $(K,g)$ and $F$. Explicit formulas are presented for some fundamental cones, including the Lorentz cone.