Does the cost of a term life insurance policy go up over time?

Yes, the cost of your term life insurance coverage may increase over time. With level term life insurance plans you pay the same level premium for the duration of your term. 

So, if you have a 10 year term life policy you would pay the same premium each year for 10 years. However, if you need life insurance to continue after the ten years ends, you would renew your policy and the premium would increase based on your age at renewal.