Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as $A\wedge_{A\wedge A} A$, like for usual commutative algebra.

My question is: can one say something about $F_{A\wedge A}(A,A)$ (function spectum), that is about topological Hochschild cohomology? Does the Gerstenhaber bracket make sense in this context? If $A$ is, say, the K-theory, does homotopy groups of its Hochschild cohomology contain some interesting elements?

Of course, it is enough to have $A_\infty$-ring structure on $A$ for this questions, but I am interested only in $E_\infty$. Besides, I am interested in spectra $F_{A\otimes S^n}(A, A)$, where $S^n$ is the $n$-sphere.