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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".