I found a new job, great. So, I quit my current job a week ago and give them a two week notice (so I'm still on this job). However, today I got a notice from the new job that they have to freeze new hire due to the company policy. What can I do? Can I ask the new company to pay me for the lost income? For how long?

I doubt you can get them to pay you for anything because most work is 'at will employment' meaning they can hire and fire you at will and you can quit without a two week notice.
Unless you signed a contract you have nothing binding with them. You might want to see if your old job would be willing to take you back. Other than that start looking for another new job. There is no such thing as job loyalty or job security any more.

You're basically scre*ed since they changed their policy. There is nothing you can do at this point unless you signed an employee contract with them. I sure hope you can take back your two week notice with the present job. Some companies let you do this and other companies say bon voyage as you leave after your two weeks are up. I'd talk it over with someone there immediately. Hopefully you turned in your resignation in a friendly manner.

Yes, begging your old place to let you stay comes to mind.

There is some legal obligation, since you acted on their promise to hire, if you have some proof that they gave you a valid offer, date to start work, hirely amount and so on, you may be able in court hold them liable for a few weeks pay ( maybe) but doing this you also destroy some of your future, since a person known to sue a place they work often finds problems latter.