Conventional wisdom says run the grade specced and printed on the oil cap as long as the vehicle is under warranty, though that's not likely to apply to an '05. It also says that unless you live in northern Maine or Minnesota, 0w will provide no real benefit over 5w.
With that said....I use whatever 0w-30 or 5w-30 synthetic is on sale and/or rebate. For that matter, I would run 5w-20 in winter without hesitation. (My Corolla is back-specced to 5w-20 though shops do not believe that since it's not in their computer.)
For normal driving and OCIs, today's synthetics or even SN/GF5 semi-synthetics let you sleep well at night, knowing your engine is protected. If you take your car to the track---not likely with an Odyssey, I reckon---or pack it past the load limit and drive to the mountains every weekend, or live in rural Minnesota and park outside, then your needs might be different.
If your vehicle starts to burn oil when you switch from 5w-20 to 0w-20 it's more likely the new brand of oil than it is the switch from 5 to 0.
Rest easy and enjoy the ride.