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Every air-conditioning area may not cause this.
In most establishments, the air conditioning system is not or could not be cleaned. With time, moulds accumulate in the duct and can cause symptoms.
Many do not allow fresh air - but re-circulate the air to save power consumption. So if there is one sick person, the virus will be spreading to everybody.
temperature itself may be responsible - but not common. Try changing the Gym. You may benefit.
Wishing speedy recovery