In the classical quickest change detection problem, the objective is to detect a change in the distribution of a sequence of random variables with minimum possible delay subject to a constraint on the false alarm rate. We consider this problem with an additional constraint on the cost of observations used  before the change occurs. This problem is encountered in many engineering application. We propose a minimax formulation for this problem. For the case when the pre- and post-change distributions are known, we show that a two-threshold generalization of the classical single threshold test is asymptotically optimal. We extend this theory to  the case when the post-change distribution is unknown.