I am making a media player app on Android and one of the big priorities is making the startup of the app quick so that the user can just start the app and click play. I want to know a good benchmark to aim for so that the user feels that it is fast, but is still within the realm of doable (given the number of calculations and networking requests I want to do on startup). I saw somewhere in the Android documentation that if it takes your app longer than 1/10th of a second to respond, it feels slow, and am wondering how that applies to startup (if people are a little more tolerant of waiting). The app pulls a significant amount of data from a website and (among other network requests) I am trying to figure out when fetching new data from the website to check for changes against the app's local cache will reach the point of diminishing returns.
five seconds is the max patience limit of an average user. As you are performing some background tasks as you suggested, it's best to display a progress bar to indicate the amount of time left for waiting. There's a lot of interesting data here for websites rather than apps, but it gives you an idea of how patient people generally are.