"Barefoot" training??

Not literally, gotta wear footwear to the gym. But, anyone try out anything really minimal for regular gym stuff? They require closed toe shoes at the fitness places, but that's all they specify. It's triple digits here, and too hot to wear shoes except when it's required. Which means taking a pair along with me. Which means hauling a bag. Having to remember not to forget my shit at the gym....

Thanks for any thoughts/recommendations. I even wear flip flops on approaches, and go barefoot, at times, on certain rock/dirt combinations. 

Best, Helen

EDIT to clarify, I'm not asking about climbing gym, but something cool enough for hot temps outside, yet allowed for using a treadmill, or that sort of thing.

I like Feiyue martial arts shoes, and I wear them with the heels flipped down when it's really hot or if I want to slip them on and off between tries/lifts.

If you get them make sure you buy from this site, because there is a French company that basically stole the name and branding and is making them as "fashion" shoes. These are the legit ones from the original company.

https://www.tigerclaw.com/feiyue-shoes-martial-arts-pr-7012.html

Look into Vivo and xero barefoot training shoes very lightweight uppers and as minimal as it gets for a training shoe that actually looks decent. I love walking barefoot and training barefoot at the park in the grass we gotta let our feet FREE!

P.S. treadmills are the worst run outside.

I'll 2nd the Xero shoes. My daily, running and lifting shoes are all Xero brand. I can feel everything through them and absolutely no ankle twisting foam. There's a 2mm sole that you can take out if you want absolutely no padding. Lots of colors styles and intended uses, minus the minimal sole they look completely normal. They have a big November sale every year (Thanksgiving/Black Friday/Cyber Monday/Anniversary).

I'll chime in! Feiyue are not the most comfortable, but they're excellent if you want a good amount of padding on the ball of your foot. They wear out quickly, but they're like a third of the price of fancy shoes. Vivo are awesome, I've had a few models and thought they were some of the best shoes I've ever had. 

Curious about the xero sandals in comparison to bedrocks, particularly on trails/slick rock.

they're dorky as all hell, but I'm a big fan of the Vibram Fivefingers. As close to literal "barefoot" as you're going to get!  I wear the V-trail Alpha for hiking and EL-X for workouts. Takes some getting used to but will develop a lot of foot/ankle strength and stability. I feel like it complements stuffing my feet into rigid climbing shoes. They make lots of options but the simpler moc styles with no laces or maybe a velcro strap have always been my favorites.