My gas furnace is approx. 30 years old. The furnace will run one cycle, shut off and approx. 15 seconds later run again for a shorter period. Everything is working properly, heating correctly etc. but runs two cycles. Any thoughts on why?

Can you tell if the gas is coming on in the short cycle? It shouldn't come on unless signaled by the thermostat. The blower has a limit switch and a relay that could be doing something funny. At least on an older furnace, the relay and limit switch are easier to trouble shoot. If it is just the blower, cleaning the limit switch might help.

If the gas is coming on, the problem is likely with the thermostat. Can't think of a good reason for to to go off and come back on again either. How consistent is this? A loose wire could cause it, but would be variable.