A $C^\infty$ diffeomorphism $\varphi$ of a manifold $M$ is cohomologically rigid if for each smooth function $f$ on $M$ there is a constant $f_0$ so that the cohomological equation $h-h\circ\varphi=f-f_0$ has a smooth solution $h$. We prove that all of the eigenvalues of the mapping on $H_1(\mathbb{T}^n,\mathbb{ R})$ induced by a cohomologically rigid diffeomorphism $\varphi$ of the torus $\mathbb{T}^n$ are roots of unity if $n<4$. The same is true for $n=4$ provided that $\varphi$ preserves orientation. We do not know whether it is true when $n=4$ and $\varphi$ reverses orientation or when $n>4$.