Air defense isn't going to help you against heavy tanks, unless you're talking about the planes that accompany them. Best defense against heavy tanks is heavy walls and compartmentalized base, since their only weakness is inability to destroy walls fast enough or shoot over them. I only ever got the dreaded heavy tank attack you speak of once and he couldn't kill the base fast enough, probably because I use walls well in my base and make sure to upgrade them. Also, keep in mind, this is not a cheap attack, the player needs to be sure his troops survive. For that reason the player attacking you will probably have all of the following going for them:
- Maxed HT for their age
- One age above you
- Brandenburg Gate (for DPS bonus)
- All HT university researches (for DPS bonus + HP)
- Germans (for DPS bonus)
Basically, that player is significantly stronger and with that kind of firepower, he could bulldoze your base with any army he chooses, he simply chose HT to save resources and retrain time. So you'd be screwed no matter what, it just adds more insult to injury knowing he has no downtime.