Apparently, grass will grow until the soil temperature is at least 1 C. If a hosta loses it’s leaves to frost but the soil temperature of the soil is greater than 1 C, will the hosta roots continue to develop?
The roots of your hosta are going to continue to develop until the ground is frozen solid. Just because the temperature is 1C the soil is still warmer than the air and everything will be fine. I continue to water my garden into December to make sure that when the soil finally freezes everything is moist.
I have included a link to Toronto Master Gardener Gardening Guide for you.
Good luck with the garden.