Audio Research VS110 Middle transformer gets very hot the 2 sides not so is this normal

 Unless the middle transformer is getting really hot I would not worry. If you can touch the top you should be OK. The middle transformer is the power transformer for both channels while the other two transformers are channel specific. Notice how much bigger it is. You can email Audio Research and ask them if they think there is a problem for peace of mind.

  The SL3s are excellent speakers. The VS110 should have no problem driving them. I drove a pair of SL3s for years with a Classic 60 and then V70. I even had them for a short while with my current Ref 110. I finally replaced them with a pair of Sonus Faber Elipsa SE speakers.

If no red plating tubes, most likely nothing to worry about.

It certainly isn't a bad design from ARC!

I used the VS110 to power Maggie 3.6R's and never had the middle transformer get that hot. Those Maggies were rated 4 ohms - with the tweeter drawing 3.3 ohms. The VS 110 drove them with ease.

Your SL3's are even more sensitive than my old Maggies - 89 compared to 86, so it's not a power handling issue. I wonder if there's an issue with the middle trans specifically? @lostbears has a point in terms of emailing ARC.

$10 infrared sensing handheld electronic thermometer.

Point it at the transformer, on different areas of the transformer, and you’ll get a reading.

From this you can guess the transformer core temperature, as you are measuring it’s surface temp (which will be different). The numbers measured will be all over the place but will have an average.

This will tell you all you need to know, in order to get the answer you seem to be asking for.

I was told a long time ago that if the power transformer gets hot but the other transformers remain cool then it's a good tube amp design.