I've read in several places that baking soda might be helpful to combat crabgrass (NOT in the lawn) but all references are about southern areas.
Will this also work in colder climates?

You could try it, and let us know if it works.

You could also use regular table salt (sprinkle on and around the weed), or vinegar (use in a spray bottle). Just dont use either of these around plants you want to keep. The salt is perfect for cracks in the sidewalk/patio. The vinegar can be used in the garden, but very carefully.

My preliminary report on the baking soda: It definitely harms the crabgrass.

After just one day (which included a rain shower), the crabgrass on the right has blackened and somehow shrunk. Before this, it was about the same size and state of health as the one on the left.

Here is another patch. It's obvious which area was treated with baking soda.

Conclusion. I would say it works, probably to the point where, if it doesn't kill the plant it might stunt it to prevent germination.

HOWEVER, it is unsightly - who wants all that white powder hanging around? - and can only be used on areas where it can't harm desirable plants and other organisms.
It is better than chemicals? In my view, yes.

In the end, as skoot_cat said, a nearby patch treated with a mix of white vinegar and salt in hot weather was completely annihilated, turning black within hours and has not seen any regrowth in two weeks. The mixture also worked on dandelions and some creeping stuff.

While the dead plants aren't exactly pleasing to the eye, vinegar and salt is better than white powder everywhere. No idea on long-term effects on the soil (because of the salt) but the area treated is not used for anything but occasionally parking a car there (actually belongs to my neighbor but she's given me free reign to abuse her weeds).

I guess, in the end, if you want environmentally friendly weed removal that looks good, the only thing that works is elbow grease. I've returned to pulling these suckers up by hand.