There are dozens of reports and complaints in the Google Web Search Help forums that the Google iOS App keeps logging users out over and over again. This can be frustrating to those who want to use personalized features such as Google Now or other saved preferences.
One user said it is even more frustrating for him because not only does it log him out, he has to go through the two-factor authentication process, each and every time, to log back in.
The bug was reported a month and a half ago and Google did not acknowledge it until a few days ago.
Jessica from Google said:
So wait it out but not using the app until it is resolved.
Forum discussion at Google Web Search Help.