Edit: Ignore completely what I wrote previously - I was going by how I normally read effect descriptions and by what I remembered from using it in WvW a long while ago, but in this case it seems to work completely differently from I would have expected. I'm not sure whether the description on this skill's tooltip is incorrect (wouldn't be the first time that happened) or whether my initial interpretation of it was just way off.
Simple enough to try out, and I finally took the time to do so. The effect lasts for 6 seconds each time it's applied, but, strangely, it seems to be re-applied every 3 seconds, starting about 3 seconds after the spirit is created. It almost seems as though the "skill" of the sun spirit has a 3 second cast time, and it keeps casting it. Hovering over its affect description while in the vicinity of a sun spirit, you can see that it keeps cycling "5s... 4s... 3s... 5s... 4s... 3s..." (i.e. 5.99s -> 3.0s countdown) until the spirit dies, then (again no idea why this is the case) about 3s after the spirit dies, you get the final 6s (5s, 4s, 3s, 2s, 1s, 0s, gone) of the effect.
While affected by the spirit's boon, I was able to inflict burning on my targets within very few hits; switching targets right after applying burning, I'm relatively certain I was able to inflict burning again within less than 10 seconds, but I could be wrong on that one. I also haven't had time to confirm with others whether the cooldown is shared, but will attempt to do so later this week (when my workload allows).
If anyone can confirm or expand on this, it would help to let me know I'm not going crazy, as the way this works is very counterintuitive from what I've come to expect about how to read skill descriptions.