Long term storage in very high or very low humidity levels will effect the moisture content in the wood in your bow. Not so much the limbs, but the riser.
Good rule of thumb is to store your bow in the same heat, cold, and humidity levels you live in.
Storing a bow in the same room as your furnace is bad news. Hanging your bow in the basement up in the ceiling joist can be really bad in the winter if your furnace is in the basement. All that heat rises and it cooks your bow & can dry it out.
Short term hot to cold, strung or unstring shouldn't make a difference on a fiberglass or carbon backed bow..... just use common sense and treat it like you would treat your family....