Let us stick to affine toric variety.

By definition, a toric variety is a variety containing a torus $T \cong (\mathbb{C}^*)^n$, with the torus action on $T$ extend to the whole variety. The torus action in the definition is the usual multiplication:

$T \times T \to T$

$(a_1,\dots,a_n) \times (b_1,\dots,b_n) \mapsto (a_1 b_1,\dots,a_n b_n)$.

But what if we allow the torus action to be any group morphism. Say

$(a_1,\dots,a_n) \times (b_1,\dots,b_n) \mapsto ({a_1}^{m_1} {b_1},\dots,{a_n}^{m_n} {b_n})$.

If we defined a category of variety similar to the definition giving above, but allowing the torus action to be any group morphism, and this group morphism extend to the variety. Do we have the SAME category of toric variety?