It would've been rather difficult for Augustus to have adjusted the number of days of August to be equal to July's when August already had 31 days and Augustus made no changes to the Julian calendar. I've never actually heard this myth you're talking about, but it doesn't make sense. The number of days in the Julian calendar derives from Caesar's commissioning of Sosigenes to fix the number of days in the Roman calendar year more precisely. Sosigenes established a 365-day year (with the provision of leap years) by extending the year by ten days, which were distributed throughout the old months to make them more equal. March, May, Quintilis (later called July), February, and October all maintained their old lengths. April, June, September, and November were lengthened by one day. January, December, and Sextilis (later called August) were lengthened by two days. February, no doubt, might have likewise been lengthened by two days (hence the placement of leap day in February), but Sosigenes did not do so, presumably for astronomical reasons that I don't know about. This is before Sextilis was called August and indeed before Quintilis was even July yet