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

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.

That can be sort of true, yes.
But peace of mind is only a $10 electronic infrared thermometer away...

I expect to hear back a number of 50C or thereabouts. Not the exact correct number, as there never really is one - but simply a range, depending on the design.