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In fact $X$ is a toric variety. "In general" the automorphism group of this variety is the torus himself. But in some cases it is not. For instance if $\varphi$ is the diagonal action of $\bf {C^*} ^n$ on $\bf C^n$ the closure of the orbit is the affine space itself and the automorphims group of this space is much bigger ; it contains the homothety group which fixes some point as remarked by Jason Starr, or in higher dimension the entire group $GL(n,\bf C)$ hence any conjugate subrgoup of the diagonal torus, and even more complicated stuff. The complete classification of toric varietes and their automorphism groups is in Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4) 3 1970 507–588.