You mean mars, a star the size of the sun will expand the green zone outward not inward, as it progresses towards the red giant status. That said, the type of star matters more than the age, since red giant is often considered seperate from our current stars status despite being a different life cycle, and larger or smaller stars have consequently different orbital needs.
With all that in mind, you must consider the fact that short of icey bodies in complex orbits that I doubt martin as a fantasy writer has time for, the world of planetos inherently possesses magical driving forces as demonstrated by the summer/winter cycle. While jupiter sized is out a super earth class planet is quite possible, despite the much larger gravity it would generate the abundance of primal era magic(As demonstrated by the fact that the valyrian freehold got into epic arcane slap fights during their conquests) means it's quite possible for the energy to have warped the nature of gravity, or if you prefer the science version, having lessened the bending of space time by propping it up with energy woven into the fabric.
Planetos follows a primal-high-low magic curve, where early magic was vast and expansive(Dragons, others, children of the forest, blood champions, etc everywhere), which was mastered during the age of empires and college into high magic, and has since collapsed into more a remnant of what it once was as the power of magic wanes and becomes the place of an elite few.