Can I raise a Boston Terrier if I don't have air conditioning in my apartment?

In addition to Shannon's suggestions, I'd also recommend looking into a cooling mat. If you give your pup a variety of options for staying comfortable, I don't imagine you'll have a problem.

When it gets hot, my Boston usually just finds a cool spot on the floor to lay down. Fans also help, but Bostons LOVE to sunbathe so good luck trying to keep them out of the sun!

As long as you don't go jogging at noon in August, or anything like that, a Boston should do fine. I'm from SoCal, so I'm familiar with the weather, and most of the places I lived didn't have a/c. You just need to take common-sense precautions -- close south-facing drapes during the day, make sure the dog has access to water, leave a fan running on really hot days, don't leave the dog in the car (even with an open window). Bostons are pretty hardy, even being smaller dogs with short noses, and conditions tolerable to you will be safe for the dog.

I've had Boston Terriers for 14 years and have only had air conditioning 5 of those years. When I didn't have air conditioning, I would make sure there was a fan on for them. At night we would open the house up and bring the cool air in, then close the house down and draw the drapes to keep it cool. My first BT loved baths, so I would leave a shallow bath drawn for him to go sit in.