IPhone 4s became hot quickly draining battery and shutdown unexpectedly

You jailbroke (is that a word? jailbreaked?) it, thereby opening the door to any application doing anything with your hardware. And now it appears that some application is doing something with your hardware. Your phone is compromised and is now trying to compromise other devices on its netblock. You're carrying a virus infection vector in your pocket.

I recommend a full restore absolutely ASAP, and I recommend really thinking about whether jailbreaking is for you.

I'm not anti-jailbreak. I just think people aren't aware of the full repercussions when they do it. Apple's not just being arbitrary in only allowing vetted software onto the devices. It's not JUST a business move on their part. It also protects users from exactly what has happened to you.

I wouldn't necessarily take Apple's word on how much of use you can get out of your iPhone given it's mostly marketing and reproduced in specific lab conditions. Here are some real world numbers from AnandTech's iPhone 5 review. The differences can be quite significant for instance, 4.55 hours on 3G and 8.19 on LTE.

Note that this is presumably for a brand new iPhone 5; if you have one that is more than a year old, it would probably be around 80-90% of it's maximum battery capacity.

Lastly, here are some ideas on maximising your iPhone's battery life:

• Disable Location and Background App Refresh for Facebook
• Disable Background App Refresh for Apps You Don't Care About
• Stop Quitting Your Apps in Multitasking
• Disable Push Email Temporarily (especially if you are using Exchange)
• Disable Push Notifications for Apps That Annoy You

It may also be worth doing the test mentioned at the top of that blog post i.e.

So here's the test: write down your usage and standby time, press the sleep/wake button (or lock button, as some call it) to put the device to sleep, and set the device down for five minutes. When you come back, take note of the change in time. If your device is sleeping properly, then the Standby time should have increased by five minutes and your Usage time by <1 minute . If your Usage time rises by more than one minute, you have a drain problem. Something is keeping your device from sleeping properly, significantly shortening the time it will last.

From my own experience, I've noticed that apps that utilise background app refresh in concert with location services tend to be pretty expensive in terms of battery usage. Dropbox and Downcast use these to upload photos and download podcasts in the background respectively - location services appear to be turned on all the time when these Dropbox/Downcast features are turned on. I turned off background app refresh and location services for both of these apps and have noticed quite significant improved battery life.