Near the cylinders it is clearly cooler
Because the end nearest the cylinder is cooled by the cylinder.
OK I did wonder about that.
Next puzzle: why is the exhaust on the RHS (shown) hotter than on the LHS?
Is the color what’s telling you where it’s hot and where it’s cold? If so, why couldn’t it be the reverse of what you are postulating – hottest near the cylinder? I’ve never measured temps here with the engine running, except of course my EGT probes, which are in a fixed location. I think some thermal imaging would be helpful to undertand the temperatures, but I’ve never seen them of a piston aircraft engine.
This is a picture of a sports car exhaust at full power and it also shows higher temp downstream of the cylinders, judging by the brighter, oranger-yellower red further downstream, even before the junction
An here’s a more conventional engine
And here’s another one with turbochargers where you can clearly see the different signature upstream vs downstream of the turbine
I reckon that exhaust is definitely Inconel 
I have the FLIR USB camera for my phone and could easily get a pic of the exhaust, but it would be only a ground run, and for obvious reasons not close up 
You would need to do it on a dyno (probably how that car engine was done) but then the temps would be different because on a dyno there is a huge blower fixed above the engine so the airflow is very different from the in-cowling airflow. Whoops I nearly gave away the answer to my puzzle…
Hint: combustion speed varies with air-fuel mixture ratio
I don’t think it is that.
It is true that on the TB20 the RHS (1,3,5) CHTs are higher than the LHS (2,4,6) ones and I believe the reason is the same one as why the RHS exhaust also breaks more often, but it isn’t the combustion product temperature because the EGTs are basically the same.
So, what could produce a hotter exhaust system with the same EGTs?
The answer can be seen in the engine photo here
This Y-shaped exhaust pipe, known variously under P/N TB20 56000009 and later TB20 56000016, is the most troublesome part of the TB20 exhaust because it combines hot gas from two cylinders
but would have been OK had Socata not lied to everybody for 22 years by saying the exhaust is made of Inconel when actually it is stainless. There is a similar pipe on the other side but that one is OK.
I would think this has a lot to do with the highly transient motion of the gases. Hot gas is only flowing out of the cylinder when the exhaust valve is open which happens once with relatively short duration every other revolution. The exhaust pipes becomes more of an organ pipe with nodes and antinodes of pressure/flow, all dependent on rpm and temperature (throttle setting). Especially in two stroke engines this is important. They won’t run properly without the exhaust pipes tuned to the engine and the rpm. It’s also important for high performance 4 stroke engines, but doubtful that Lycoming/Continental has given it much thought, and with a turbo there are probably other considerations?
Don’t know how relevant it is, but at node and anti-nodes, the relative oscillations in flow is zero and max respectively. This will for sure make the transfer of gas temperature to the pipe different, but how much, and in which direction?
Peter wrote:
is the most troublesome part of the TB20 exhaust because it combines hot gas from two cylinders
Which is almost exactly where this section blew out on my Bonanza A36 (IO550)
