Nanoparticles have a small finite size. Do the still obey the third law of thermodynamics?
Can a temperature be assigned to them at all?
It really depends on the environement. If your nanoparticle is alone in a cold vacuum, it will undergo radiative cooling up to $T=0$.
If it is inside a huge thermostat with a certain temperature, then it will have the same temperature due to energy exchange with the termostat.
Microscopic scale energy density fluctuations happen even in an infinite system, thus a thermometer must be sufficiently macroscopic to avearge those fluctuations.
Above I implied that the temperature is low enough in order to speak of your nanoparticle. If the temperature is high enough, then your nanoparticle may melt or evaporate. You see, in Physics many things are implied. Matematically these things are expressed with some inequalities.