If you aren’t looking for extreme experiences, better to travel to Mongolia in June or period from August to October. During that time temperatures are doable for any average mortal.
Consider however traveling in July if you want to attend largest national festival in Mongolia — Nadaam. It’s held annually between 11–13 July in Ulaanbaatar. Mongolians from whole country gather in the capital to participate in „three mannish competitions”. These are: crazy horse racing, traditional wrestling and archery (women participate too).