As most of you know, Captain America uses his shield to protect himself from a lot of stuffs, be it physical attacks, projectiles, explosions, electrical shock, fire, cold, ...
Thanks to the properties of the main alloy used in his shield (aka vibranium), most of the time, he comes out unscratched.
However, the vibranium is a metal. And metal is mainly know for being good conductors, both electrical and heat, whether cold or hot. Despite this, the shield is oftenly depicted blocking off electrical and heat based attacks, and the shield holder doesn't seem to feel it.
Black Panther blocking off Hoarfen's cold breath during their assault on Asguard with Captain America's shield, The Avengers: Earth's Mightiest Heroes, season 1 episode 26 "A Day Unlike Any Other"
During the above scene, Black Panther was standing inside Hoarfen's mouth and used Captain America's shield to protect himself from the cold breath, and he came out undamaged, although his hands should have frozen from holding the shield.
Some answers to this Quora question claim that vibranium is indeed a conductor in some way (though they're talking only about electrical conductivity). But then, Captain was depicted blocking a blow from Mjollnir without getting hurt.
Captain America blocking a blow from Mjollnir using his shield, The Avengers: Earth's Mightiest Heroes, season 1 episode 9 "Living Legend"
Is there any canon information about the conductivity of Captain's shield?
This question might be related, although it's speaking of the adamantium alloy.