Deformable image registration is fundamental for many medical image analyses. A key obstacle for accurate image registration is the variations in image appearance. Recently, deep learning-based registration methods (DLRs), using deep neural networks, have computational efficiency that is several orders of magnitude greater than traditional optimization-based registration methods (ORs). A major drawback, however, of DLRs is a disregard for the target-pair-specific optimization that is inherent in ORs and instead they rely on a globally optimized network that is trained with a set of training samples to achieve faster registration. Thus, DLRs inherently have degraded ability to adapt to appearance variations and perform poorly, compared to ORs, when image pairs (fixed/moving images) have large differences in appearance. Hence, we propose an Appearance Adjustment Network (AAN) where we leverage anatomy edges, through an anatomy-constrained loss function, to generate an anatomy-preserving appearance transformation. We designed the AAN so that it can be readily inserted into a wide range of DLRs, to reduce the appearance differences between the fixed and moving images. Our AAN and DLRs network can be trained cooperatively in an unsupervised and end-to-end manner. We evaluated our AAN with two widely used DLRs - Voxelmorph (VM) and FAst IMage registration (FAIM) - on three public 3D brain magnetic resonance (MR) image datasets - IBSR18, Mindboggle101, and LPBA40. The results show that DLRs, using the AAN, improved performance and achieved higher results than state-of-the-art ORs.