The local push setting looks like the screenshot below. You can find it in App Configuration under the first row, your server settings, then under Internal URL.
Turn off “Local Push” and it’ll stop running in the background.
I understand you wish to closely manage your battery usage and I want you to feel empowered to control it, but since others may stumble across this post unsure about what’s happening, I am going to soliloquize a bit about what’s going on. tl;dr: the way it’s reported makes it look long in ‘activity’ but it should be low in ‘usage’.
Pinging roughly every minute is the system-suggested schedule for Local Push connections, and the app does it using a method which allows the system to optimize the exact timing. I believe the system rounds up this usage to a full minute, hence the large ‘activity’ time, so keep in mind that it’s not actually burning through battery that entire duration. For example, in your screenshot, the Home Assistant app running for ‘5.5 hours’ consumed less battery than Brave running for 5 minutes. This isn’t too surprising because using the display actively is probably the largest possible battery cost on the device. This is the expected experience of reported usage from testers and myself during the beta period, and has held strong through release.
Battery usage is one of my primary concerns for the app’s various features. For example, app sacrifices some hyper-exact location accuracy in favor of focusing on in-or-out-of-Zone status, which drastically reduces battery usage for background location. Local Push is taking some new amount of battery (hence why there’s an option for it), but it’s also absorbing some existing battery cost for some kinds of notifications which require downloading images, videos, etc.
I believe it’s worth the net-new battery cost (as a default) because of the speed of notifications, reduction in rate limiting, and in general the Home Assistant philosophy around local control - we should aim to get things to you phone without going through servers on the internet.