I want all the timestamps of every time that an app has started running.
I tried searching in the Play Store but I didn't find one. I downloaded an app called AppUsage which shows the Last Launched timestamp but I want all the timestamps
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There's LaunchStats which should work as is on versions up to (including) Android 4.0 Ice Cream Sandwich. This app scans the system logs for all app starts since the device was booted and extracts the data you want.
However starting with 4.1 Jelly Bean the necessary READ_LOGS permission has been restricted to system apps due to security concerns. It is still possible, but you need to grant that manually via the developer console on your PC (adb is part of the Android SDK):
adb shell pm grant de.goddchen.android.launchstats android.permission.READ_LOGS
The Android system stores statistics on this, which on most devices can be accessed via a hidden menu:
You should see a list of your apps, which tells for each app how often it was started, and how long it ran. Using the drop-down box on top, the list can be sorted by usage time, usage count, or app name.
Remark: AFAIK this list will be reset when you restart your device.
Edit: In case your device provides no dialer (thanks @ce4 for pointing out my lapsus), it will require some searching to find the appropriate menu, but it's doable. Some launchers (e.g. Apex Launcher) provide the possibility to create shortcuts to "activities" (rawly: screens provided by apps). Long-tap a free space on your homescreen, then select "shortcuts", then "activities". Now scroll through the quite long list to find it. On my LG Optimus 4X the activity is called "Test" and has a "Gear" icon (same as the "Settings" app has in this menu), so that's the first place to look for: scroll to "Settings", open the drop-down, look for "Test".