This is an interesting question indeed. I can not offer a full answer,
only the following observation.

First consider the case $n=3$, i.e., $T_3$ is the maximal torus of
$\operatorname{PGL}_3$ with the natural action of $S_3$. Let's use the
Hochschild-Serre spectral sequence to compute
$\operatorname{H}^\bullet(S_3,T_3)$.

First, let $G=\mathbb{Z}/3$ and use the standard resolution
$$
\cdots\to \mathbb{Z}[G]\stackrel{\Delta}{\rightarrow}
\mathbb{Z}[G]\stackrel{t-1}{\rightarrow}\mathbb{Z}[G]\to\mathbb{Z}\to 0
$$
where $t$ is a generator of $G$ and $\Delta=1+t+t^2$.
Now
$\operatorname{H}^\bullet(G,T_3)=\operatorname{H}^\bullet(\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}[G],T_3))$
is the cohomology of the complex
$$
T_3\stackrel{(t-1)^\ast}{\rightarrow}
T_3\stackrel{\Delta^\ast}{\rightarrow} T_3\to \cdots
$$

To evaluate this, describe the action of $G$ on $T_3$. The maximal
torus $T_3$ in $\operatorname{PGL}_3$ is given by diagonal matrices
$\operatorname{diag}(a,b,c)$ modulo scalar matrices $\operatorname{diag}(a,a,a)$. Using
representatives $\operatorname{diag}(1,a,b)$, the action of $t$ on $T_3$ is
given by $\operatorname{diag}(1,a,b)\to \operatorname{diag}(b,1,a)\sim
\operatorname{diag}(1,b^{-1},ab^{-1})$. Consequently, the action of $t-1$ on
$T_3$ is $(a,b)\mapsto ((ab)^{-1},ab^{-2})$ and the action of $\Delta$
on $T_3$ is trivial. From this, we obtain
$$
\operatorname{H}^i(G,T_3)=\left\{\begin{array}{ll}
\mu_3 & i \textrm{ even}\\
k^\times/(k^\times)^3 & i \textrm{ odd}
\end{array}\right.
$$
where $\mu_3$ is the group of third roots of unity in $k$.

Next, we describe the action of $\mathbb{Z}/2$ on
$\operatorname{H}^i(G,T_3)$. On $T_3$, the non-trivial element $\sigma$ acts as
$\operatorname{diag}(1,a,b)\mapsto \operatorname{diag}(1,b,a)$. It follows, that $\sigma$
acts via $x\mapsto x^{-1}$ on $\operatorname{H}^i(G,T_3)$. Again, we use the
standard resolution to
compute the cohomology $\operatorname{H}^j(\mathbb{Z}/2,\operatorname{H}^i(G,T_3))$. The
element $\sigma-1$ acts as identity on $\mu_3$, and the element
$\sigma+1$ acts trivially on $\mu_3$. Similarly, $\sigma-1$ acts as
identity on $k^\times/(k^\times)^3$, and $\sigma+
1$ acts trivially. The result is that
$$
\operatorname{H}^i(\mathbb{Z}/2,\operatorname{H}^j(\mathbb{Z}/3,T_3))=0
$$
for all $i,j$, so the Hochschild-Serre spectral sequence degenerates
and shows that $\operatorname{H}^i(S_3,T_3)=0$ for all $i$.

I claim that the same argument shows that the $p$-torsion in
$\operatorname{H}^i(S_p,T_p)$ is trivial. In fact, we only need to compute the
cohomology of the normalizer $N_p$ of the $p$-Sylow with
$T_p$-coefficients. This is done as above via the Hochschild-Serre
spectral sequence. In the first step,
$$
\operatorname{H}^i(\mathbb{Z}/p,T_p)\cong\left\{\begin{array}{ll}
\mu_p & i \textrm{ even}\\
k^\times/(k^\times)^p & i \textrm{ odd}.
\end{array}\right.
$$
On these groups, the quotient
$N_p/(\mathbb{Z}/p)\cong\mathbb{Z}/(p-1)$ acts via cyclic permutation
of the powers. As above, the map $\tau-1$ is an isomorphism in the
Hom-complex, showing the vanishing of cohomology
$\operatorname{H}^\bullet(N_p,T_p)$ as above.

So, as a partial answer to your question: the cohomology in case $n$ prime is only annihilated by $n$ if it is completely trivial.

I would love to know the general statement, what is $\operatorname{H}^\bullet(S_n,T_n)$, and I would expect it to be known and written somewhere. A seemingly related question I would
also like to see answered: what
is a reference for cohomology of the normalizer of a maximal torus in
an algebraic group. The above cohomology groups are part of the
Hochschild-Serre spectral sequence computing the normalizer of the
maximal torus in $\operatorname{PGL}_n$. Over algebraically closed fields, the
homology of the normalizer of the maximal torus (with finite
coefficients) can be understood in terms of \'etale
cohomology, see the discussion in Chapter 5 of K. Knudson ``Homology
of linear groups''. Can somebody take it from here?