In Physics (Thermodynamics), there are 2 kinds of quantity: extensive and intensive. The extensive one like volume, mass, energy, entropy, particle number...are additive. therefore their sum and average are physically meaningful. Intensive one like temperature, pressure, chemical potential... represent a quality of the system. They are not additive. If you want to average or sum temperatures of 2 systems, say, to average or sum the temperature of your hot tub and the wall of your bathroom, this so-called averaged temperature or total temperature is no longer temperature. It just becomes an abstract number. In real world we have temperature of your hot tub, we have temperature of your cold bathroom wall, but we don't physical thing that corresponding to that averaged temperature or total temperature calculated by averaging the temperature of hot tub and wall.
To summarize, average temperature is a abstract number, a quantity not defined in physics.