Are there any downsides to leaving personal hotspot perpetually on?
It does drain your battery; whether it's worth it for you to turn it off is up to you and how long your battery last. 

Also something worth noting is that by leaving it on and walking around other people can see your hotspot and might try to connect. Even if they're unsuccessful it will drain your battery even faster. If they are successful who knows what they might do while connected. 

In general though, any app on your phone that sends/receive data is going to drain your battery more than an app that doesn't.