I am making a mobile application and currently I have a custom action bar with just text describing the functionality of the current page and a back button to return the user to the previous page. I am trying to promote my product through the logo but it might be forced if I include it in the action bar. Would including the logo in the action bar be forced and just add extra confusion?
By the terminology you use, I believe you are referring to Android, so let me focus on this platform first.
Action bar recommendations were, back then, to display the app icon: https://developer.android.com/design/patterns/actionbar.html
App icon The app icon establishes your app's identity. It can be replaced with a different logo or branding if you wish. Important: If the app is currently not displaying the top-level screen, be sure to display the Up caret to the left of the app icon, so the user can navigate up the hierarchy. For more discussion of Up navigation, see the Navigation pattern
This approach, however, has changed with the introduction of Material Design: https://material.io/guidelines/layout/structure.html
The app bar, formerly known as the action bar in Android, is a special kind of toolbar that’s used for branding, navigation, search, and actions.
The nav icon at the left side of the app bar can be:
A control to open a navigation drawer. An up arrow for navigating upward through your app’s hierarchy. Omitted entirely if no navigation is required from this screen. The title in the app bar reflects the current page. It can be an app title, page title, or a page filter.
Icons on the right side of the app bar are app-related actions. The menu icon opens the overflow menu, which contains secondary actions and menu items like help, settings, and feedback.
For iOS, I think it is also uncommon, however apps like Medium, Trello or Skype do include it in the top bar - but I believe it is rather due to branding than something that would allow the recognition in the future.
Of course, opinions may vary but I think it will add more clutter and confusion cost than the app recognition gain, so it is obsolete. If a User has already downloaded the app, it is enough for him/her to see it in the launcher.
However, there is one case in which I think it would be valuable. Let us say this is an Android app that is not visible anywhere in the launcher (e.g. a keyboard settings pane, often removed from the launcher screen and only left in the drawer). Or something that is triggered strictly from a widget, or notification, without an actual app icon. Then it may be valuable to show it. Still, I believe these are really rare occasions.