Let $P$ be a smooth connected projective variety (say, over complex numbers); $H$ is its smooth hyperplane section. What can be said about the Zariski cohomology of $H$ with constant coefficients? It is certainly zero in positive degrees if $H$ is irreducible. Does it vanish in lower degrees (those that are smaller than $\operatorname{dim}P-1$) in general?

Upd. It seems that with rational coefficients the answer is positive. Indeed, the cohomology groups in question would be exactly the weight zero part of the singular cohomology of $H$ (considered as a sequence of mixed Hodge structures), whereas for $i<\operatorname{dim}P-1$ the Weak Lefschetz theorem yields that $H^i_{sing}(H)$ is pure of weight $i$. Yet for torsion (or integral) coefficients the answer is not clear to me. Could the Zariski cohomology in question have torsion? Is there an upper bound on the exponent of this torsion (that depends on $\operatorname{dim}P$)?

P.S. Cf. Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?