When sending as manifest cargo 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 and other regular cargo products are unaffected by this pricing changes.
During the covid pandemic, airlines have globally raised their cargo rates several times due to required revenue takings to make that route operational and make up for the shortfall of lack of fare paying passengers. With Hong Kong this is especially so due to mandatory quarantine and passenger numbers down by over 98% or more. With this, airline have raised their cargo rates significantly and with little warning, stop accepting pets altogether, or as in most cases, stopped flying from/to Hong Kong altogether.
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.