In Halo Infinite’s pause menu, the screen is blurred but the game image still used to be rendered in 4K resolution, even if the player couldn’t see it.
The team Used the Hive Mind and chose to lower the resolution of the pause menu to reduce power consumption without impacting player experience.
The whole process is detailed in their blog post: Seven Steps that Made Halo Infinite More Energy Efficient and How Your Studio Can Do It Too.
If you have designed your pause screen to not require any background movement, you can take this one step further and stop rendering the game view altogether. Instead, cache a single render texture once and then just show that for as long as the pause is active, rather than rendering a new identical background image every frame. In some games, this could save up to 50% of your energy use while paused.