Even without going to a gym, there are lots of activities you can do without going to the gym - for example biking and hiking. There are many opportunities that will taken you into the wonderful fresh air of Tuscany. Gym in Greve, I found this one: Asd Double Step, Palestre e Fitness , Viale Vittorio Veneto, 76, 50022 GREVE IN CHIANTI (FI) - TOSCANA, +39 334 2447744
While the end of July/start of August can be quite warm, I agree with Donna and suggest spending time outdoors rather than inside. You might need to do it earlier in the morning than planned if you do get warm days, but you'll enjoy the wonderful views and vineyards.
From Greve, we've taken a walk to Montefioralle and then taking the white roads (unpaved roads that connect the farm houses/wine estates) to get to San Cresci. From there, made our way across to Badia a Passignano. Gorgeous views!!
This is just one example.... another is to go up the roads towards Castello di Uzzano and follow CAI signs (red and white hiking markers) up towards Rugliana. You make a loop and head back down towards Greve.
Another walk is from the Castello di Mugnana to La Panca, again along the CAI paths, then opposite the valley to Castello di Cintoia and Cintoia and back to the the Castello di Mugnana.
The entire area is perfect for these types of walks, which I personally prefer over the gym They are all moderate-difficult as there are hills (but not super steep, aside from the one from Greve to Montefioralle).