What can I do? You are wise.

Thanks for the compliment... In the early 60's I was already building my own Crystal radio when I was 8 years old... Never really left electronics behind me...
You should check these things:
First of all, is your operator AT&T, T-Mobile or???
Then check out their network on internet like I did with AT&T. Are you in any of the cities living where 4G/LTE coverage is available?
If not, that rules out that you are having the same problem what I discovered. If that is the case, I guess some heavy apps can cause this if they refresh often. Permanent on line like with push email usually doesn't require so much power as frequently (re)initiating a connection. An app can cause this...
If you look at internet, things like "bing bar" like to follow where you are surfing. They log all movement which requires them to have open connections between them and you. An app can cause the same behaviour... On mobile phones that also means (more expensive) traffic...
That is why I really don't like apps... and bing bar for that matter...
Looking at how many times this "Hot" subject returns in Google, I guess that for most of you that situation of 4G/LTE is true... Already some people feel it has something to do with 4G/LTE...
If you type in Google:
Samsung hot battery or
iPhone hot battery
you see the same complaint. Check this post out also:
iPhone 4 is running hot and battery draining faster than normal - Neowin Forums
This guy was just near the border of Mexico and he experienced a lot of phone registerring also... There goes the battery...
I quote: "* Yesterday I was around the border and my phone kept connecting to a Mexico cell tower. And it was like...stuck on that damn tower even when I was quite a ways away and should have been well within range of a state side AT&T tower. I kept my phone off hoping to avoid roaming data charges (I suspect I'll be calling AT&T on that one). It was peculiar all the same. Kept connecting to "movistar". Maybe my phone got hacked by a drug cartel."
Again the simple word is transmission networks...
Cells for mobile is still cheapest to build, but with high density networks in cities the cells get closer together and handovers between one and the other cell makes your phone do a lot of "useless talking".
Much better are satellite networks. Just 32 of them to cover every corner of the planet. And you have a direct line of sight. Nothing blocks the signal...
It is never the screen or something else which drains the battery; it is transmission which does it...
Operators have a big challange on their hands. All of us are walking with one phone and some of them carry a batallion with them... All these phones want to register at a cell. Cells have a limitation on the number of registerred phones.
Fortunately you phone can cover some distance to a neighboring cell... But the power of your phone needs to dynamically "pump up the wattage" to be able to register at a nearby cell...
All this directly influences the duration on batteries...
Complicated matter, this radio technology...
If you are in a 4G/LTE cell, just check what happens if you make a daytrip out of that region towards a 3G area...
Fred