One-shot object detection (OSOD) uses a query patch to identify the same category of object in a target image. As the OSOD setting, the target images are required to contain the object category of the query patch, and the image styles (domains) of the query patch and target images are always similar. However, in practical application, the above requirements are not commonly satisfied. Therefore, we propose a new problem namely Cross-Domain Object Search (CDOS), where the object categories of the query patch and target image are decoupled, and the image styles between them may also be significantly different. For this problem, we develop a new method, which incorporates both foreground-background contrastive learning heads and a domain-generalized feature augmentation technique. This makes our method effectively handle the object category gap and domain distribution gap, between the query patch and target image in the training and testing datasets. We further build a new benchmark for the proposed CDOS problem, on which our method shows significant performance improvements over the comparison methods.