  An irreducible algebraic variety $X$ is rigid if it admits no nontrivial
action of the additive group of the ground field. We prove that the
automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique
maximal torus $\mathbb{T}$. If the grading on the algebra of regular functions
$\mathbb{K}[X]$ defined by the action of $\mathbb{T}$ is pointed, the group
$\text{Aut}(X)$ is a finite extension of $\mathbb{T}$. As an application, we
describe the automorphism group of a rigid trinomial affine hypersurface and
find all isomorphisms between such hypersurfaces.
