  Let $\lambda$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For
any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $\lambda$ with
measure-theoretic entropy equal to topological entropy, we show that the set of
points with nondense orbits is hyperplane absolute winning (HAW). This
generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of
the circle.
