When a white light beam is passed through a prism do the resulting colors differ in temperature?

Answer 1

Colors do not have a temperature associated with them.
For example, “Blue” objects can have a higher or lower temperature than “Red” objects depending on the situation.

Light carriers energy in the form of discrete packets called “photons”. The amount of energy which is contained in one of these photons depends on the frequency (or wavelength) of the light.
Energy = h * v
Energy = h * c / lambda
(Where h is Planck’s constant)
These two equivalent equations for the energy contained by a photon of light show that it is directly related to the frequency of the light (v), and inversely related to the wavelength of the light (lambda). Frequency = the speed of light / Wavelength.

The wavelengths of light which we associate with the different colors possess different amounts of energy per photon of light.

“Red” light, which is located at one end of the visible electromagnetic spectrum, has a wavelength of around 700 nanometers.
“Violet’ light, which is located at the other end of the electromagnetic spectrum, has a wavelength of about 400 nanometers.
The difference in the two’s wavelengths means that the “violet” light (which the higher frequency and lower wavelength) will contain a greater amount of energy per photon than the “red” light.

For light of wavelength 400 nm, each photon possesses about 3.10 electron Volts worth of energy (an electron volt, eV, is a very small unit of energy equal to 1.602 E-19 Joules).
For light of wavelength 700 nm, each photon possesses an energy of about 1.77 eV.
The energy difference between the two ends of the electromagnetic spectrum is about 1.33 eV.

Answer 2

Light doesn't have temperature. Different color light is emitted by objects of different temperatures, and the heat transfered to an object by different colors of light would be greater for higher frequencies, or energy levels at equal intensity, but light itself isn't hot or cold.
The higher frequency colors like blue do have more energy content than the red ones, if that's what you mean.