Two Points on Opposite Sides of the Earth with the Same Temperature.

Question

I've often heard the fact that there are two diametrically opposed points on the surface of the Earth with the same temperature and barometric pressure. If you dug a hole through the Earth, the temperature and barometric pressure would be the same as the place you left. But then I thought, "It takes time to dig through the Earth. Temperatures and pressures can change over time. Is there always a pair of points with the temperature and barometric pressure of point 1 before you start digging being the same as the temperature and barometric pressure at point 2 after, regardless of how long it takes to dig?" I soon realized the answer was no, as the temperature at every point on the Earth could rise above the highest temperature before digging began. But if we fix the global average temperature and pressure, does the answer change? What if you fix the global average temperature and the maximum and minimums to stay the same? Is there a pair that is the same before and after the digging? It seems to me like instead of mapping the surface of one sphere to the plane, you are mapping the surface of two half-spheres onto it. Any solutions to any of the questions would be helpful.