  This paper can be seen as an update to part of the author's dissertation. We
study the mod $p$ cohomology of the pro-$p$ Iwahori subgroups $I$ of
$\operatorname{SL}_{n}(\mathbb Q_{p})$ (and
$\operatorname{GL}_{n}(\mathbb{Q}_{p})$) for $n=2$ and $n=3$. Here we use the
spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p})
\Longrightarrow H^{s+t}(I,\mathbb{F}_{p})$ due to Sorensen, and we do explicit
calculations with an ordered basis of $I$, which gives us a basis of
$\mathfrak{g} = \mathbb{F}_{p} \otimes_{\mathbb{F}_{p}[\pi]} \operatorname{gr}
I$ that we use to calculate $H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$. We note
that the multiplicative spectral sequence $E_{1}^{s,t} =
H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$ collapses on the first page by noticing
that all maps on each page are necessarily trivial, and this allows us to
describe the above group cohomology groups and all cup products. Finally we
note some connections to cohomology of central division algebras over
$\mathbb{Q}_{p}$ and point out some future research directions.
