Earth-s-Axial-Tilt-The-Specifics
...xnip...In the course of a year. We have different seasons because of this. However, the locations of the north and south poles are not static. In reality, the axial tilt of the Earth oscillates between 22.1° and 24.5°. The Earth’s axis also wobbles around itself, which accounts for the shifting axial/obliquity angle. It results from the gravitational pull of the Sun, Moon, and other planets. However over time, Earth’s environment and climate have been significantly shaped by its axial tilt...xnip...
I suspect climate change is also affected when our solar system goes above or below the protective dust shielding along Galactic plane and we are all of a sudden subjected to all sorts of radiation from the stars and black holes at the center of the galaxy that cause an increase in lightning in thunderstorms, but that is for another thread!
The tilt does indeed change, over time. But the change is quite slow. The full cycle of this change in tilt is about 41,000 years.
Any space elevator system would change right along with the tilt of the earth.