I use Chrome as my default browser and it syncs to my desktop. I just looked at Settings>Battery and Chrome is not listed as a top user. I then looked in GSam Battery Monitor Pro and it listed Chrome at 0.1% of my battery usage. Fwiw, my battery is at 60% having been on for 15 hrs, 26 min. It's the original Razr, not the maxx. I use google everything just about - gmail, calendar, contacts, chrome, music, etc. and my google acct is set to sync everything. So when I open up Chrome on my Razr, I can go right to any open tabs on my pc, use my logins, etc.
Just to say my experience is all positive with Chrome and with syncing with my pc.