The thermodynamic definition of temperature has been found to be emergent from the underlying particulate nature of matter. It is connected with an average over the kinetic energy of individual particles.
Here v is velocity of a molecule, m its mass, k_B the Bolzman constant and T the temperature
The kinetic energy requires to have a degree of freedom, which is fine in gases. In solids the degrees of freedom are the rotations and vibrations of the molecules, as the molecules themselves are bound and thus do not have degrees of freedom in space. The same for the internal constituents of molecules, atoms , etc. They exist in a bound state and a temperature cannot be defined for them. Their only contribution comes into contributing to the mass of the molecules.
One can stretch the definition by using the kinetic energy of a particle in the formula, and derive a temperature. All one is saying is that "this would be the temperature of an ensemble of particles that have this kinetic energy on average"
Another stretch of definitions is found here.
Thus at the subatomic level there does not exist a temperature for the bound quarks and gluons as no kinetic degree of freedom exists.
In the comment the quark matter subject has been broached. This is a hypothetical state of matter where the energies are such that the QCD asymptotic freedom behavior emerges. This can happen in two ways :
1) during the Big Bang ,
The earliest phases of the Big Bang are subject to much speculation. In the most common models the universe was filled homogeneously and isotropically with an incredibly high energy density and huge temperatures and pressures and was very rapidly expanding and cooling. Approximately 10−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the universe grew exponentially. After inflation stopped, the universe consisted of a quark–gluon plasma, as well as all other elementary particles.
The temperatures here are defined by the kinetic energies of the hypothesized particles and it is supplied by the energy of the universe as it evolves after the Big Bang
2) and is searched for in ion ion collisions at LHC.
In these heavy-ion collisions the hundreds of protons and neutrons in two such nuclei smash into one another at energies of upwards of a few trillion electronvolts each. This forms a miniscule fireball in which everything “melts” into a quark-gluon plasma.
The temperature here is defined by the kinetic energy of quarks and gluons in the plasma that have degrees of freedom as for a while they are asymptotically free. The energy is supplied by the accelerator.