what is the best way to clean a rubber TT record mat.

Brown Bear Car Wash. Oh sorry, Luxman? Not Audi?

I would remove it to the sink, run some warm water over it while gently brushing with a clean microfiber cloth. If the water runs off nice and clean then you are done. If not then rinse again with distilled water and air dry. 

Yeah... soap and water. Unless it's cracking and dried out and useless, refrain from using chemicals.

Don’t use dishwashing soap. It contains degreasing agents which remove the substances which make the rubber pliable. As Millercarbon said, if a cleaner is required, use car wash soap, only very diluted, and be sure to rinse all residue off after cleaning. I would avoid car wash soap that contains finish protectants (claim to be be “wash and wax”). Meguiars Hi Tech wash or other paint shop friendly products contain no silicones that can leave undesirable residues behind. 

I’ve cleaned my Garrard rubbers for years in mild soapy water in a basin without any ill effects as I can feel or see. Now that I think about it, it is an organic detergent. I found that water alone didn’t clean fingerprints etc.
Edit: Just spent 5 minutes flopping the rubber and it's still as supple as ever.

Baby shampoo

I didn't have any mild soap so I just used warm water and patted the mat down. Looks a lot cleaner. Thanks. I was curious how much the Luxman mat was and WoW! the 5mm x 29cm rubber TT mat costs $345 at Acoustic Sounds. A lot more than that felt mat I used on my former Linn TT.

How about a "sticky" cat hair roller....No????

Hi,
green soap and lukewarm tap water, then rinse with distilled water, dry well with microfiber, nothing more needed.