Caching is an effective mechanism for reducing bandwidth usage and alleviating server load. However, the use of caching entails a compromise between content freshness and refresh cost. An excessive refresh allows a high degree of content freshness at a greater cost of system resource. Conversely, a deficient refresh inhibits content freshness but saves the cost of resource usages. To address the freshness-cost problem, we formulate the refresh scheduling problem with a generic cost model and use this cost model to determine an optimal refresh frequency that gives the best tradeoff between refresh cost and content freshness. We prove the existence and uniqueness of an optimal refresh frequency under the assumptions that the arrival of content update is Poisson and the age-related cost monotonically increases with decreasing freshness. In addition, we provide an analytic comparison of system performance under fixed refresh scheduling and random refresh scheduling, showing that with the same average refresh frequency two refresh schedulings are mathematically equivalent in terms of the long-run average cost.