Developer Yaroslav Brovin has a tutorial up for customizing the color of a TListBox selected item in Delphi XE7 Firemonkey. It shows you how to use the Bitmap Style Designer to export the style bitmap and then edit it in the image editor of your choice. There are some graphics which show you exactly which part of the style image you should change to modify the color of the selected TListBoxItem (there are actually 4 images you must update to handle all of the resolutions). Once you’ve updated the images you can load them back into the Bitmap Style Designer and then save out your new style. The style that you save out can then be loaded in the TStyleBook. It appears like you will need to customize the style for each platform (Android, IOS, OSX, Windows) as each one will be different. This tutorial write up is posted as an answer to a question on StackOverflow. You should also be able to use this same functionality in Appmethod. I think this same technique also applies to customizing a TListView.
Head over and check out the full question and answer on customizing the TListBox selected item color over on StackOverflow.