How does the apparent path of the sun change as the earth revolves around the it?
The entire Sun-system, including the Earth, orbits around the center
of our galaxy, Milky Way. The galaxy's gravitational centripetal force
keeps the solar system, as a whole, in an orbit around its center, at a
distance of about 27 K light years, with period 250 M years, nearly.
The contribution ( if any ) by the Earth alone, for changes (if any ) in
the path of the Sun, is negligible.