In fact, just like in the case of non-equivariant algebraic $K$-theory, there is a homotopy-invariance for equivariant $K$-theory (of nonsingular varieties).  Addressing your question specifically, for any action of an algebraic group $G$ on affine space, the answer is:
$$ K^G_i({\Bbb A}^n) = R(G) \otimes K_i(k) $$,
where $k$ is the ground field and $R(G) = K^G_0(k)$ is the representation ring.  See Theorem 4.1 of Thomason's foundational paper "Algebraic K-theory of group scheme actions".