That's exactly what I do.
Those people who have investigated such things say that temperatures of more than 30 degrees damage the rope but anything less is fine. The evidence also points to detergents and softener doing no damage as such. However, the detergent also has no real cleaning benefit because it is designed to remove grease, oil etc, rather than straight-forward mud and grit.
There are some people who use washing machines who think it prudent to give it a run empty first to remove any residual washing powder. Personally, given the above, I don't.
Finally, it's also good practice to store rope in a cool, dark place. Personally I hang them from small lengths of old rope in the (blacked out) garage.
For more info, have a look at this. It's written by the eminent Dave Elliot (so must be right!)
http://web.ukonline.co.uk/members/andy.mccarron/cncc/cor.htm