I find early game DoWs to be the biggest challenge (and the only real challenge) in Apollo. Basically any AI will DoW -- I got creamed by PAU in a recent game, figured I was safe since he was friendly and he's the cooperative sort, but bought two tiles in one of my cities near him and he instantly DoW'd. Then creamed me because my army was on the other side of the map defending against aliens, and there was a frikkin canyon wall separating most of my settlements.
While AIs have trouble taking each others' cities, that's because they both always have big armies and defensive buildings -- whereas your cities will fall like flies to a handful of tier 1-2 units. Early game AI wars are especially detrimental on MP, since your human opponent will be spamming colonies and TRs and getting way ahead while you're wasting resources on military.
If you are near a lot of AIs in a closed space, get defense structures and might 1 to make your units last longer with instaheals. If you're really concerned, beeline for affinity 4 and your first unique unit -- these crush all non affinity units and once you have a couple the threat of AI warfare pretty much vanishes. Do not count on being friendly with your neighbors to avoid wars -- they will DoW anyway unless you can actually sign a co-op agreement. They also seem less willing to be paid to attack each other than in civ5, but maybe I've just had bad luck with that.