It is cumulative, as I understand it, if you're in more than one war at the same time. If, for example, I'm at war with Persia and England and my war weariness level is at a 2 with Persia and a 5 with England (no clue on the numbers), my overall war weariness is 7.
If I make peace with Persia, the effective weariness drops to 5, though. At least, that's the best I've seen it explained that matches any in-game experience.
The war weariness with Persia is still there, it's just not effective and not affecting my citizens, as I understand it. It's latent, for lack of a better word.
Continuing the example, if 2 turns later, I declare war on Persia again, my cumulative war weariness will probably jump to 8 or 9, as the 2 latent weariness from Persia is still there and returns to effect. Add 1 or 2 for declaring.
Arathorn
P.S. I repeat, all numbers are purely for example purposes! I have no idea how weariness is stored and at what threshholds it starts affecting the people.