The War Temple holds the Divine Powers responsible for improving the warfare power of your city.
Every upgrade of the War Temple will unlock some new Divine Powers to be researched (including access to higher tiers of units). To be ale to conduct research in the Temples, you need to get enough Elixir. That you can get either by generating it in the Sanctuary, or from the Shrines which randomly show up on the map after killing beasts.
The War Temple itself does not provide any special bonus other than providing a place to research Divine Powers and unlock more powerful army units.
* Total Power amount provided on each level