Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the local cohomology module $\mathrm{H}^i_{\mathfrak{m}}(R)$ is of finite length. If we write $\ell(\cdot)$ for length of a module of finite length, I am interested to know whether it is true that $$\ell(F(\mathrm{H}^i_{\mathfrak{m}}(R)))=p^d\cdot\ell(\mathrm{H}^i_{\mathfrak{m}}(R))?$$ The trivial case is when $\mathrm{H}^i_{\mathfrak{m}}(R)=0$. Any results from the literature in special cases where this formula holds, or any example in which the formula does not hold would be interesting. I don't know of any special case or example, except for the trivial case.

Let $p=2$, $R=k[[x,y]]/(x^2,xy)$. Then $H^0_m(R) \cong R/m= k$. $F(k) = R/m^{[2]}= k[[x,y]]/(x^2,xy,y^2)$ has length $3 \neq 2\times 1$. In general for a finite length module $M$, the condition that $\ell(F(M))=p^d\ell(M)$ is pretty restrictive. For $M=k$ this forces $R$ to be regular (Kunz). Also, when $R$ is a complete intersection such condition forces $M$ to have finite projective dimension, see Claudia Miller's survey. 

