I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. There also exists some fixed function $f$ on natural numbers, such that for any $m\geq f(n)$ we have $H^m(X, \Gamma(n))=0$ for smooth varieties $X$ over the base field. I am interested in examples other than the motivic cohomology.
Rational examples are also fine, meaning having $\Gamma(0) = \mathbb{Q}$ instead of $\Gamma(0) = \mathbb{Z}$.
