An affine toric variety is defined as a variety $X$ that contains a torus as an open dense subset such that the natural action of torus on itself extends to an action on $X$.
I would imagine that the torus action extends to all of $X$ just becuase the torus is embedded in $X$, but that portion of the definition always seems to be stated explicitly as a hypothesis. Are there examples of varieties $X$ that contain an embedded torus where the action doesn't extend? And what goes wrong in the usual construction of a cone from such a variety and torus embedding?