Opposite sides of the universe have the same temperature, yet according to the standard big bang theory these points are too far apart for light to have traveled from one side to the other in the age of the universe; that is, they cannot have exchanged heat to even out their temperature. why, then, do they have the same temperature?

At the beginning, particles and radiation used to have the same temperature as they were in a state of thermal equilibrium where they interacted with one another.  However, when the Big Bang occurred, they evened out their temperature and a rapid inflation of the universe carried them far apart keeping the same temperature.

answer; first the above statement is correct and the answer you are looking for is (energized); ///good luck