Let $k$ be a field of characteristics $p$ and $R$ be any ring where $p$ is not invertible.

Asuume that $F:Var_{k}\to D(R-mod)$ is a cohomology theory of smooth algebraic varieties over the field $k$ with values in $R$-modules that satisfies etale descent and has $F^0(\mathrm{Spec}\,k)=R[0]$ such that for any geometrically connected variety $X$ the structure morphism induces an isomorphism $R=F^0(\mathrm{Spec}\,k)\simeq F^0(X)$(these assumptions are satisfied for crystalline cohomology with torsion-free coefficients, e.g. over $W(\bar{k})$). Then non-vanishing of $F^1(\mathbb{A}^1_k)$ is forced.

Indeed, consider the etale Artin-Schreier $\mathbb{Z}/p$-cover given by $\mathbb{A}^1\to\mathbb{A}^1,t\mapsto t^p-t$ (a generator of the cyclic group acts by $t\mapsto t+1$). The Hoschschild-Serre spectral sequence looks like$$E_2^{i,j}=H^i(\mathbb{Z}/p,F^j(\mathbb{A}^1))\Rightarrow F^{i+j}(\mathbb{A}^1)$$
If $F^1(\mathbb{A}^1)=0$ then it induces an isomorphism(there are no differentials that could touch this term) $H^1(\mathbb{Z}/p,F^0(\mathbb{A}^1))=E^{1,0}_2\simeq F^1(\mathbb{A}^1)=0$. But we've assumed that $F^0(\mathbb{A}^1)=R$ with any automorphism acting trivially, so $H^1(\mathbb{Z}/p,F^0(\mathbb{A}^1))=R/p\neq 0$.