A pair I see a lot are "heating days" and "cooling days." What are these?

Heating days and cooling days refer to use of furnace/boiler or A/C. In NYC a landlord is required by law between Oct 1 and April 20th to provide heat any day the temperture falls below 55º at night.

if there is a late (or early) 'cold snap' (a cold night in late August--when the tempurature drops to 48 degrees- ) and the landlord doesn't have the boiler/furnace ready to provide heat (and doesn't have to!) He can not be fined. but come Oct, if there is no heat, he risks being fined in housing court, (a housing violation, get enough, and the city puts a lein on the property)

if Oct is unseasonable mild.. (and night time temps remain above 55º--) the land lord does not have to provide heat! (so a 58º--pretty cool) might result in a chilly apartment in the AM--no over night heat!--and again, no fine to landlord.

NYS has some energy rebate programs for the elderly, and some of these are also based on 'heating'/'cooling days'--
i don't particapate in these programs (STAR is the name of one as i recall) but these programs also use 'heating'/'cooling' day info.