Current carrying capacity of 28 AWG wire?
I am assuming you are going to use only one twisted pair (out of the 34 available) for 1 A. In that case, you should be fine for your application using 28 AWG copper assuming that your cable is surrounded by open air at 30 °C or cooler. The first link's recommendation is close to the (United States) National Electric Code formula. Note that the fine print for the second link says that the power transmission values are "very very conservative."

If more than one twisted pair is carrying a significant amount of current or the cable will not allow heat dissipation, use multiple pairs to carry your current.