According to the Farmer's Almanac, 'Early arrival of crickets on the hearth means an early winter'. No one has shown a direct correlation between a greater number than usual making for a harder winter (except that it will mean a hard time for a greater number of crickets).
https://askinglot.com/what-does-a-lot-of-crickets-mean
You can tell the temperature from the rate of cricket chirps:
To convert cricket chirps to degrees Fahrenheit, count number of chirps in 14 seconds then add 40 to get temperature.
Example: 30 chirps + 40 = 70° F
To convert cricket chirps to degrees Celsius, count number of chirps in 25 seconds, divide by 3, then add 4 to get temperature.
Example: 48 chirps /(divided by) 3 + 4 = 20° C
http://www.almanac.com/cricket-chirps-temperature-thermometer
Folk wisdom
Here are the 20 Signs of A Hard Winter.
Thicker than normal corn husks
Woodpeckers sharing a tree
Early arrival of the Snowy owl
Early departure of geese and ducks
Early migration of the Monarch butterfly
Thick hair on the nape (back) of the cow's neck
Heavy and numerous fogs during August
Raccoons with thick tails and bright bands
Mice eating ravenously into the home
Early arrival of crickets on the hearth
Spiders spinning larger than usual webs and entering the house in great numbers
Pigs gathering sticks
Insects marching a bee line rather than meandering
Early seclusion of bees within the hive
Unusual abundance of acorns
Muskrats burrowing holes high on the river bank
"See how high the hornet's nest, 'twill tell how high the snow will rest"
Narrow orange band in the middle of the Woollybear caterpillar warns of heavy snow; fat and fuzzy caterpillars presage bitter cold
The squirrel gathers nuts early to fortify against a hard winter
Frequent halos or rings around sun or moon forecast numerous snow falls.
https://www.farmersalmanac.com/20-signs-of-a-hard-winter-479
