Correct solutions were submitted by Paul Lee, Burkart Venzke and Bill Webb.
Let f(t) be the temperature t years before now. Then f is continuous with f(0)=f(2006). Set g(t):=f(t+1)-f(t) and note that g is also continuous. If we show that there is a t with g(t)=0 then t and t+1 years ago form the desired time pair.
Consider the values g(0), g(1), ..., g(2005). The sum of these values telescopes to f(2006)-f(0)=0. If one of these values is zero, we are done. Otherwise (at least) one of these values is positive and one is negative. Then we are done by the Intermediate Value Theorem.