New ESA and cost of living.

Unfortunately not because New style ESA isn't a means tested benefit and doesn't qualify you for any of the payments. You will still receive £66/67 per month between now and March if you're named on an energy account.

Full details and eligibilty criteria for all the CoL payments are here. https://www.gov.uk/guidance/cost-of-living-payment

Have you also checked entitlement to Universal Credit? (Although it's too late now to be eligible for any of the CoL payments) Your New style ESA will be deducted in full from any UC entitlement. If you live with a partner you need to claim as a couple and your joint circumstances will be taken into consideration.

If you have savings/capital of more than £16,000 you're excluded from claiming. Use a benefits calculator to check entitlement. https://www.entitledto.co.uk/benefits-calculator/Intro/Home?cid=21e02bec-0ab6-4d4a-b2fb-b5762dae9eeb