The Cargo rate is same all year around for pets so the flight cost will be the same to travel in what are thought to be “peak times” like School Holidays and Christmas time. Although your ticket price for personal flights can double at peak times, pets travelling Cargo are unaffected by this pricing changes.
The only way pet flight prices would rise would be if the airline is increasing their cost , can happen on an approximately annual basis.
Pet travel agents do charge surcharges for weekends and Public Holidays and night times for assistance with clearances or delivery, and some airport government customs departments do close at nights or weekends depending on the Airport.