Conifold has provided a very good answer, so he/she should be credited for it.
I'll just add the following clarification. Your question contains two parts:
- Every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point (Borsuk-Ulam theorem).
- The Borsuk-Ulam theorem is applicable to the earth and its temperature.
The first statement can be considered to be a priori knowledge as it does not depend on empirical investigation to determine its truth. The second statement does require empirical research (e.g. is the earth a sphere, in euclidean space,...), hence, if true, would be a posteriori (empirical) knowledge.
In general, mathematics, without observation, does not say anything about the world.