Is the temperature of the universe rising?
There is a difference between the "temperature of the universe" and the temperature of the cosmic microwave background radiation (CMBR).
The former can be changed by physical processes going on in the universe and for example, the conversion of gravitational potential energy, or the release of nuclear binding energy, into the thermal energy of particles. The CMBR temperature on the other hand is fixed when it is formed and modified just by the expansion history of the universe; it represents the temperature of a blackbody radiator with the same spectrum as the CMBR.
In the study you refer to, the "temperature of the universe" is the density-weighted mean electron temperature, and is of order $10^6$ K. These electrons have been heated via a variety of physical processes, ultimately linked to the formation of clusters of galaxies, galaxies and stars (for example supernovae, or collisionless shock heating in gravitationally accelerated flows - Kravtsov & Yepes 2000; Bykov et al. 2008), and have cooling times that are long compared with the age of the universe.
In contrast, the CMBR spectrum was formed about 400,000 years after the big bang, was essentially fixed at that point (at around 3000 K), and is only modified subsequently by the universe's expansion history, which stretches the wavelengths leading to a cooling temperature (currently 2.7 K).
The two temperatures would have been similar around the epoch when the CMBR was formed but have diverged since then because the matter became effectively transparent to the CMBR and decoupled. According to the paper that is referred to in the question, the density-weighted mean electron temperature has increased by about a factor of 3 between $z=1$ and the present day; from $7\times 10^5$ K to $2\times 10^6$ K. Over the same period, the CMBR would have cooled from 5.4 K to 2.7 K.
This is a helpful review:
A new study by an international team of researchers, including members of the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), suggests that the mean temperature of gas in large structures of the Universe has increased about 3 times in the last 8 billion years, to reach about two million Kelvin today.
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The study determined that about eight billion years ago (at a redshift z=1), the mean electron temperature was some 700,000 Kelvin, rising to about two million Kelvin today. Furthermore, the scientists determined that its evolution is almost entirely driven by the growth of structures, as gas is shock heated in collapsing large-scale structures.
Since it is a fact that the cosmic microwave radiation is a few kelvins , it must be obvious that one is talking of the gas around large structures. The usual temperature taken in timelines of the Big bang is the cosmic microwave radiation temperature.
So it is two different "temperatures of the universe", as far as I can understand, the electron gas temperature not directly connected to the temperatures used for describing the Big Bang expansion , and the cooling of the cosmic microwave radiation with time.
I hope that an astrophysicist answers to make it clear.
You can define quite a few "universe temperatures":
- Cosmic microwave background (2.7K as of now)
- Cosmic neutrino background (theoretical, but lower than CMB, probably like 2K)
- Cosmic gravitational waves background (theoretical, but more or less obvious, lower than the above)
- Temperature of the baryon matter in some region of the universe (way higher, generally 10^2..10^7K)
All these things are in one way or another "decoupled" from each other - i.e. they cannot exchange significant amount of energy over observable timespan, so they retain their different temperatures. (2) and (4) can even host more than one particle populations, decoupled from each other and having different temperatures.
The article in question deals with the temperature of the electrons around the large cosmic structures.
Keep in mind that the temperature of the other more or less ordinary things in the same region may be quite different. If you have some kind of dust in the same place, it will be quite possibly near the CMB temperature. The gas is simply not dense enough to allow for an efficient heat exchange.