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The concept of a temperature $T$ is routinely used in discussing the thermodynamics of the early Universe. It enters the discussion via equilibrium phase space distribution of various particles that comprise the thermal bath.

Intuitively, it refers to the temperature of the thermal plasma at any point in time in the early universe. I think, it should be thought of as one common temperature $T$ shared by all those particle species that are in equilibrium. When some species decouple (say, the cosmic photons after the last scattering), its temperature evolves differently than the plasma as a whole.

Is this intuitive picture sufficient to carry on with the study of the thermodynamics of the early universe or do we need to be more precise and mathematical?

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Your intuitive picture is correct. An example of a decoupled species is the cosmic neutrino background which consists of neutrinos which decoupled from the rest of the universe when the universe was approximately one second old. These neutrinos have an estimated temperature of $1.95$ K, whereas the photons in the cosmic microwave background, which decoupled hundreds of thousands of years later, have a temperature of about $2.7$ K.

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