You palm should be fine as long as you live in a Zone 9 or higher. Phoenix Roebilinii is hardy to 24F, but leaves begin to show damage at 30F. Just keep your palm as dry as possible. Do NOT cut off the brown or fertilize in the winter because it will only stress your plant. You can continue covering it but it is not that important. Your palm should recover as soon as Spring as long as you fertilize and water as needed.

I have a robillini palm in a large pot with potting soil and kept it indoors this winter, it did fine until I changed out the potting soil for a sand/potting soil/vermiculite mixture. The new mixture hold water really well. The tree is now brown and drying out with leaves dying. The new shoots out the top are becing dry also. There is a white poweder type at the base of the new shoots...is there any thing I can do?