Infinity gate / M.R. Carey.

The Pandominion: a political and trading alliance of a million worlds - except that they're really just the one world, Earth, in many different realities. And when an AI threat arises that could destroy everything the Pandominion has built, they'll eradicate it by whatever means necessary, no matter the cost to human life. Scientist Hadiz Tambuwal is looking for a solution to her own Earth's environmental collapse when she stumbles across the secret of inter-dimensional travel. It could save everyone on her dying planet, but now she's walked into the middle of a war on a scale she never dreamed of. And she needs to choose a side before it kills her.