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. If $G$ is a (split) torus, the representation ring is just the group ring of the character group; choosing a basis $x_1,\ldots,x_n$ of characters identifies $R(G)$ with the Laurent polynomial ring ${\Bbb Z}[x_1^\pm,\ldots,x_n^\pm]$.
(The general statement is that when $f\colon X \to Y$ is a $G$-equivariant affine bundle, there is an isomorphism $$f^*\colon G^G_*(Y) \to G^G_*(X),$$ where $G^G_*$ denotes equivariant $K$-theory of coherent sheaves. For nonsingular varieties $X$ one has $G^G_*(X) = K^G_*(X)$. See Theorem 4.1 of Thomason's foundational paper "Algebraic K-theory of group scheme actions".)