What is the "temperature" of the galaxy? The temperature of a gas is a measure of internal energy density. If the atoms have a high kinetic energy, the temperature is high.
Can we do the same thing with the galaxy, where the stars are like atoms?
If so, how uniform is the temperature?
I would guess that the galaxy is rather cold. It looks like the stars have "condensed" into spiral arms like water droplets in a cloud. Is there such a phase transition?
Since the galaxy started as diffuse dust and gas that collapsed into individual stars, why would the stars be so hot and yet the galaxy is cold? This is the opposite of a hot gas of atoms in the ground state.
 A: One can definitely do thermodynamics and statistical mechanics on systems of orbiting point masses. See Galactic Dynamics by James Binney and Scott Tremaine for a through treatment. Basically, one can calculate the evolution of the probability distribution of velocities at different points over time.
The problem is that the system is not in thermal equilibrium: there can be subpopulations with very different velocity distributions in the same region. One example is the thin disk stars (fairly young ones) and the thick disk (older): the thick disk has much broader velocity distribution (dispersion) than the thin disk, and hence higher temperature. Stellar clusters often show velocity dispersions that don't neatly fit a single temperature.
Why is the stars hot and the galaxy cold? If you think of stars as a monoatomic gas the kinetic energy per kelvin per mole of stars is $3R/2=12.47$ J/K. Plugging in a typical kinetic energy of $10^{38}$ J you get an absurdly high temperature - but this is mostly because we use the wrong scale. The real answer is that being able to radiate away energy as radiation is the reason the galaxy could condense, the stars could condense, and gas in between could cool down. Of course, stars are also heating up because of fusion, but the potential energy released by the condensation has been vast.
Peebles and Fukugita estimate that $10^{-4.9}$ of the mass-energy of the universe is due to dissipative gravitational condensation, while the galactic mass is about $1.5\cdot 10^{-3}$ - about 1/120 of the mass-energy of the stars have been released this way, about the same that can ever be released by fusion (!)
