In an email, Marcia asked me if she should quit keeping her yeast in the fridge. She doesn't make bread very often so she keeps it in there. It still foams when she begins to hydrate the yeast at the beginning by mixing it with water and sugar. She wonders if it is a mistake to keep it cold.
The short answer is you can keep it cool in the fridge. That is no mistake.
I got the long answer by going out to google and asking "Why do people keep their yeast in the fridge?", even discovering a scientific explanation about the membrane around yeast and how hydration happens and where best it happens. I don't think any of us need to know all of this about yeast.
Here is my practical advice. I buy my yeast in the large can at Costco. I never keep it in the fridge, but in a quart jar on my shelves. I probably do that because it keeps well for me there and my fridge space is more valuable kitchen real estate for me than my cupboards. I might have to buy that large tin about 3 times a year. Like you, Mary keeps her yeast in the fridge. The instruction say keep in a cool place. It is fine in the freezer. It is fine in the fridge. It is probably fine, as mine is, on my shelves as I am making bread on a routine basis, only buying it when I run out and don't have time to make it.
I am not suggesting a person should use this.
My mother could no longer find yeast cakes easily
and had to switch to what was easily available.
I think the bottom line is ... you were asking if keeping the yeast cold will inhibit the rising of your dough. Once the yeast gets bubbling / foaming -- it has long forgotten whether it was kept frozen or cold or on the shelf. It is working for you and will make your bread rise.