Here's the place I'm taking my cat this week:
www.ppkdoha.net
I've taken him here every summer and other holidays for 4 years. I think it's one of the only places that has an indoor space AND outdoor space for each cat. (most only are a metal cage)
They charge 45QR a day + 5QR day if it's summer (for the A/C)...
It's way out of town, so if you are short on time they pickup or drop off in town for 100QR.
There are several other places but I'm not sure of their websites...