Honestly for the men in the group, running without tunics on may feel better at the time, but in the long run under these conditions it is not healthy. I would suggest a light tunic of fabric, to soak up the sweat and yet still keep the skin cool. Too much evaporation from your sweat pores will cause faster dehydration. To the sad looks from the slaves, sorry, but health is paramount during the summer.
I wear a silk shirt during the heat, soaks up the sweat yet still keeps the skin damp and dries faster than a cotton or wool type tunic and is lighter so any breeze will keep the skin cool.
Men, keep this in mind. And yes, I do know what silk signifies on a man.
Raz