In interactive content search through comparisons, a user searching for a target object in a database is asked to select the object most similar to her target from a small list of objects. A new object list is then presented to the user based on her earlier selections. 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 target objects are selected from a non-uniform probability distribution. We also assume that objects are embedded in a doubling metric space which is fully observable to the search algorithm. Based on these assumptions, we devise an efficient comparison-based search algorithm whose cost in terms of the number of queries can be bounded by the doubling constant of the embedding c, and the entropy of demand distribution, H. More precisely, we show that the average search costs scales $CF = O(c^5 H)$, which improves upon the previously best known bound and is order optimal for constant c.