Let $G$ be a (profinite) group. It is known that if $H^n(G_p,A) = 0$ for all $p$, $S_p$ the Sylow subgroups of $G$, then $H^n(G,A) = 0$.
Are there other localglobal principles for different sets of subgroups?
Let $G$ be a (profinite) group. It is known that if $H^n(G_p,A) = 0$ for all $p$, $S_p$ the Sylow subgroups of $G$, then $H^n(G,A) = 0$. Are there other localglobal principles for different sets of subgroups? 


The localglobal principle you are citing comes from the fact that for any open subgroup $H\leq G$, $H^n(G,A)\stackrel{\text{Res}}{\longrightarrow}H^n(H,A)\stackrel{\text{Cor}}{\longrightarrow}H^n(G,A)$ is multiplication by $[G:H]$. So from that you can derive lots of localglobal principles. E.g. as a generalisation of the one you cite, you can deduce that if $H_1$ and $H_2$ are two open subgroups of coprime index such that $H^n(H_i,A)=0$ for $i=1,2$, then $H^n(G,A)=0$. 

