Is unary language with polynomial power context sensitive?
Not true for linear polynomials.  E.g let $p(n)$ be $3n+2$, then L is generated by the regular grammar  'S --> aaaS | aa'.  [unless you mean 'at most context-sensitive', not 'exactly context-sensitive']

The Pumping Lemmas for both regular and context-free grammars seem to imply that any language over a 1-letter alphabet can be decomposed into a (not necessarily disjoint) union of sub-languages corresponding to linear polynomials.  If the set of linear polynomials in question can be made finite (which I suspect must be true, but I can't prove or disprove off the top of my head [and I am not sure it even matters]), then any higher-degree polynomial must take on values not achieved by any polynomial in the linear set, in which case such languages must be at least context-sensitive.

Also: A strategy for building a context-sensitive grammar for a polynomial language over one alphabet is to construct a sequence of $k$ sub-grammars, each of which takes $m$ consecutive $a$'s and transforms it into $b_km+c_k$ consecutive $a$s for some constants $b_k$, $c_k$ [think "Horner's Method"] and then 'cascades' to the next sub-grammar.

So if you mean 'at most context-sensitive', then yes.