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If you by "cone" mean exactly that $A$ should be isomorpic to $\mathrm{gr}_{\mathfrak m}A$ it seems that the following is counterexample: Let $G=\mathbb G_m$, $A=k[x,y,z]/(x^2+y^3+z^5)$ with $tx=t^{15}x$, $ty=t^{10}y$ and $tz=t^{15}z$ (exponents chosen more or less at random). Then the tangent cone at the origin (the fixed point) has affine algebra $k[x,y,z]/(x^2)$ and hence is not isomorphic to $A$. This is just raising your cusp example one dimension so that it becomes normal.