Android - Why does my hotspot stop after a period of time?

You most likely have a hotspot timeout setting enabled.

You can disable it by going to your Advanced Hotspot Settings, and editing the timeout value to never.

Note that in some versions of Android in order to access the Advanced Hotspot Settings you need to first access the WiFi Hotspot Settings and then press the phone's Menu button.

Just a note for others out there, the setting is not very easy to find (at least on my Samsung Galaxy S3 mini Android 4.1.1).

Go to Wireless and network -> More settings... -> Tethering and portable hotspot -> Portable WiFi hotspot. Once there, you'll have to press the Menu button (bottom left of your phone) and up pops Timeout settings!

One can only guess why they decided to hide that setting in such a hard-to-find place....