Which organs in the body are responsible for measuring temperature?
Human thermo detection comes from a protein, expressed in high levels in the nervous system.
As per comments, there are a group of transmembrane proteins called transient receptor potential cation channel subfamily V (TRPV) ion channels that can detect temperature changes. Although the exact mechanism of the protein remains unclear, a study published in Nature 2010 showed that a protein called TRPV1 opens and closes to allow increased or decreased sensitivity to temperature change.
There is no specific organ that is responsible for temperature detection, however TRPV1 is expressed in high levels in the nervous system.