It depends on the outage, If Cox's equipment is down as well, meaning their batteries failed or otherwise isn't working then you will not come back online.
HOWEVER, if others in your neighborhood have backup power and their internet IS working, then I would have your generator checked and ensure that the sine wave is good. If you are out of phase, a lot of electrical equipment will not function properly. Your modem could be one of those. If so, you can fix by getting a UPS, and putting your modem and other sensitive electronics on it. I have an APC Smart 1500 and a BR24BPG additional battery so my server and modem always has reliable power. Larger and better UPS's cost a few extra dollars, but the equipment they protect, well, that speaks for itself.
It depends the scope of the power outage. There is a lot of infrastructure between you and the internet. All of it requires power. Some of it have their own backup but that only last so long. If your looking to not lose your internet during a power outage, look into a router with dual WAN and get a cellular backup. Comcast already offers something like this called "Storm Ready Wi-Fi" Maybe Cox will come out with something similar now that they offer cell service. I know they already do for Cox Business.