There are varying opinions on this … Some professionals say it doesn't matter, so long as you do not mow when the ground is very wet, as this can cause damage to the turf, and also rutting if you have a riding mower. If you are flexible there is an optimal moment for lawn mowing
However, there are some pros that say mid-morning this the optimum time of day to mow the lawn. The logic behind this is that the lawn needs time to heal before evening, that the freshly shorn grass blades are susceptible to fungus and disease and really need the benefit of the day to dry and heal before evening sets in. So, if you have the luxury of choosing when to mow in the day, there is a case that mid-morning is the optimal time of the day.
Now, more importantly make sure that your blades are sharp, as this will affect the lawn more so than the time of day that you mow. IF your blades are not sharp then the lawn is torn as the dull blades hits it, this causes damage to your turf and makes it more susceptible to disease and damage.
Keep in mind, if you are going to be working with a professional service you will probably have to pay a premium if you are wanting to have the lawn mowed at specific time of day. These professionals make their schedules out by route density. Meaning they complete their stops in the most efficient manner possible as they navigate their way throughout the neighborhood, town, or city. So asking for a specific time will be invasive to their schedule, and you will probably have to pay extra. Take for example a St. Augustine turf that grows in Florida, it would be extremely hard for a lawn service provider to bag and haul all those clippings off.
The Street Smart Gardener recommends keeping your lawn mowing blades higher during the summer heat.