I have also used app token in my application. I have saved it to SharedPrefs with Private mode as:
final SharedPreferences prefs = getSharedPreferences(
"app token", Context.MODE_PRIVATE);
When you save a sharedPref with Private mode, it can only be accessed by the calling application (or all applications sharing the same user ID). And whenever user logs out, clear all the sharedPrefs, so that, there will not be any copy of your app token saved in local storage of the device. This should be safe to use, and security will be ensured.
Update:
App token should not be stored in SharedPreference since if the device is rooted, the sharedPrefs file can be exposed. AccountManager can be used to store token but again it can also be exposed if device is rooted. Best way is to encrypt a token with a strong encryption algo and then save it into AccountManager. The key used for encryption should be unique with every user and device combination so that even if key is compromised, hacker would be able to retrieve only data for that specific user and not all.