In comparison-based active learning, a user searching for a target object navigates through a database in the following manner. The user is asked to select the object most similar to her target from small list of objects. A new object list is then presented to the user based on her earlier selection. This process is repeated until the target is included in the list presented, at which point the search terminates. We study this problem under the scenario of heterogeneous demand, where objects are requested with different frequencies. Whether the search algorithm has access to the embedding of objects or not, we provide two algorithms for deciding which sequence of objects to show to the user and bound their costs in number of queries. Our bounds relate the cost of content search to two important properties of the demand distribution, namely its entropy and its doubling constant. These illustrate interesting connections between content search through comparisons to classic results from information theory.