Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}\_p$ as outlined in Krause-Nikolaus.
We use the fact that $\mathbb{F}\_p$ is initial $E\_2$ ring with $0=p$ to compute


$$\mathbb{F}\_p \otimes\_{\mathbb{S}} \mathbb{F}\_p \cong \mathbb{F}\_p[{\Omega^2 S^3}]$$


Then,


$$THC(\mathbb{F}\_p) \cong \mathrm{Hom}\_{\mathbb{F}\_p[{\Omega^2 S^3}]}(\mathbb{F}\_p,\mathbb{F}\_p)$$


Now using


$$\mathrm{colim}\_{BG} {G} = \{ \star\}$$


We have that


$$\mathrm{colim}\_{\Omega S^3} {\mathbb{F}\_p[{\Omega^2 S^3}]} = \mathbb{F}\_p$$


and hence,


$$THC(\mathbb{F}\_p) \cong \lim\_{\Omega S^3}{\mathbb{F}\_p} \cong \mathbb{F}\_p^{\Omega^2 S^3}$$


where the action is trivial because $\mathbb{F}\_p$ is discrete. Assuming the above is correct, I now am not sure how to compute this.



What is the topological Hochschild cohomology of $\mathbb{F}_p$?
Let me write $HH^S(B) = THH(B) = B \wedge\_{B^e} B$ for topological Hochschild homology, and $HH\_S(B) = F\_{B^e}(B, B)$ for topological Hochschild cohomology, where $B^e = B \wedge\_S B^{op}$. For $B$ commutative the $B^e$-module action on $B$ factors through $\mu : B^e \to B$, so by adjunction we have $F\_{B^e}(B, B) \cong F\_B(B \wedge\_{B^e} B, B)$. Hence $HH\_S(B) \cong F\_B(HH^S(B), B)$. For $B = H\mathbb{F}\_p$ we have $\pi\_\* HH^S(B) = HH^S\_\*(B) = \mathbb{F}\_p[x]$ with $|x|=2$
a primitively generated Hopf algebra, so dually $HH\_S^{-\*}(B) = \pi\_\* HH\_S(B) = Hom\_{\mathbb{F}\_p}(\mathbb{F}\_p[x], \mathbb{F}\_p) = \Gamma\_{\mathbb{F}\_p}(\xi)$ is a divided power algebra with $\xi$ dual to $x$.


Closer to the approach you outline: The final lim calculates the mod $p$ cohomology of $\Omega S^3$, which is the Hopf algebra dual to the mod $p$ homology of $\Omega S^3$. Either one can be calculated with the Wang sequence, or Serre spectral sequence, or by reference to the James construction.


