the only thing wrong with August is possible heat. We were there in August of 2003 when it was 105F for about a week and didn't cool down much at night. We had an apartment with excellent cross draft, but it didn't matter -- it was miserable. All the usual tourist sites are up and running and you won't even notice the few restaurants or businesses that take August vacations.
We don't usually travel in summer for this reason; for years when we were working we traveled in May. If traveling in summer absolutely get a hotel with AC and read the reviews -- some hotels turn off the AC at night or have very weak AC. This is iMHO essential in August.
If you experience a heat wave then you do the things you do anywhere to cool down -- go swim in a pool, visit museums with AC (we spent 3 full days in the Louvre which even then had uneven AC but was cooler than outside.)
Most buildings with AC in France cool only slightly -- you don't get the refrigerator cooling you find in the US, but lowering the temperature to 80F from 95F is still a life saver. With global climate change, weather is less predictable and Paris is getting warmer.