The answer is anything you please. It's your world, and gods typically have the power to be... well.. anything you want them to be.
Typically we give them fears that us mortals have, so that they resonate with us more.
If you want to have some fun, though, you can break out the Lövheim cube of emotion
The cube of emotion is a recent model which theorizes that there is a connection between our emotions and the three monoamine neurotransmitters. While still an early model that needs to be defended more, it is the first to truly tie emotions to chemicals. As such, it permits us to use neurochemistry to explore new sides of emotion.
Fear appears in the bottom left corner. It's where you have high dopamine but low noradrenaline and low serotonin. Now your gods might not have neurotransmitters, but we can look at their behavior and see if those neurotransmitters might give you some ideas. What follows is a gross oversimplification of how our brain works, but it's interesting enough for me:
- Noradrenaline is associated with unpredictability in the environment. If we're having trouble with predicting the environment, noradrenaline will spike. In fear, its low, so we're pretty sure the environment isn't actually changing all that unpredictably.
- Seratonin is associated with how happy the body is. When you eat food, seratonin goes up to let you know you did a good job. It's the link to your lizard brain. If seratonin goes low, that means your lower brain is pretty certain that things are bad.
- Dopamine is associated with reward seeking behavior. High dopamine means you think there are rewards to be had (or in the case of fear, you think there might be a way out, you just have to find it)
So when you put these together, you see that in fear, your more base self is quite certain things are bad, and you're not seeing them changing, and you're pretty sure you need to seek a way out.
Let's put this against an example. Arachnophobia. First, an inspirational photograph:
Something deep rooted in your lizard brain has taught you that this pattern of legs and eyes is bad. Really bad. Bad bad bad bad news. It lets you know "you are failing at life right now. This is not how it is supposed to be." Worse, things aren't changing. Your husband isn't coming by to whisk the spider away. In fact, the soon-to-be-about-to-sleep-on-the-couch-for-a-week fool is laughing at you. (In his defense, it is a cute little jumping spider!). Your body checks the scene again... yep, it's not changing. There is nothing to make this spider go away for you.
But you know there's a solution. Somewhere in this god forsaken spider-infested world, you know that you can make it better. Of course, you have to get past the nerves first. Perhaps a haiku will do. I composed my wife a haiku for such a moment a while back. I called it "Spider," and it went something like this:
Spider: splat splat splat splat,
splat splat splat splat splat splat splat,
splat splat splat splat splat!*
I think I captured the art of the hakiu pretty solidly in that one. It was missing a cutting word, though. Mostly spatting-words. Oh well.
So if you take those three attributes of fear, an unhappy base self, lack of change in the environment regarding that unhappiness, and the idea that you might be able to do something about it, you have a pretty generalized sense of what one might be fearful of. If your god has some underlying essense that can be happy, you're on the road towards the god being able to have a fear.
May I suggest your god have a fear of spiders? Just sayin'
* The original poem ended with the line "splat splat splat, all gone!" but my wife was insistent that wasting two syllables worth of splats was not worth it, just in case the spider might have pulled through, so the poem is shown here in its more refined form.