What is the actual temperature of a quasar? My daughter asked me last night: What is the hottest thing in the universe?

The first thing that sprang to mind was the core of a fresh neutron star (this question agrees: What is the hottest thing in the universe?), so I told her that and then tried to verify it on the Internet. Unfortunately, most of the articles online claimed that the hottest things in the universe are quasars, since they have a "brightness temperature" of 10 trillion kelvin. That sounded fishy, and a review of the literature revealed that brightness temperature has no direct relationship to actual temperature.

Poor journalistic practice aside, what is our best estimate of the temperature of these objects?
A question/answer over at physics.se  discusses the issue of quasar spectral output.  Part of one answer says,


  The percentage of radiation that is thermal versus non-thermal varies
  from quasar to quasar, and for a single quasar is going to be
  different at different wavelengths, so there isn't one simple answer
  (and also it depends on what angle you are viewing the AGN from
  relative to the jet axis)
  
  And similarly, there isn't a single value you can quote for the
  accretion disk temperature. Different AGN have different black hole
  masses, different matter infall rates and generally different
  environments. And a single accretion disk doesn't have a single
  temperature either: the temperature is higher towards the center, but
  to the details of what the emission spectrum looks like depends a lot
  on what parameters you put in to the model and what assumptions you
  make about energy transfer within the disk etc.


So it's not really accurate to estimate quasar thermal temperature from the "Brightness temperature," because the latter only applies to objects closely fitting Planck's black body law.