I mean, you could just convert the Farenheit or Celsius degrees to radians like they were angle degrees. "Bake at 6.109 radians for 45 minutes" still can mean "Bake at 350 degrees for 45 minutes" if you accept the implicit Farenheit scale. Radians would still be ambiguous regarding the base scale used, but it's as ambiguous as "degrees" is so not really an issue.
So I mean, there's no real reason to do it but also no reason you can't.
Joules is unfortunately a vector because it's over a distance in a direction. Temperature is a scalar. Sometimes scalars are better than vectors.
Edit:Ok for those who don't actually understand joules in its units J=KG•M2/s2 or N•D, it's force which is a vector over a distance, this requires a magnitude and direction. This is because force is a vector and Joules is using force. All of you are starting to be confidently incorrect... Joules is a vector you can search it up.
My scale for expressing mean kinetic energy flux is superior to your scale for expressing mean kinetic energy flux. I have formed an identity around this and will smugly argue about it on the internet.