Cheap digital thermostat turns off furnace when cold?
99% sure this will be a thermostat that needs batteries. A fairly common approach to getting "smart" features without changing from old two-wire (heat only) mechanical thermostat wiring. I've lived with several variants - the batteries and method to get to the batteries can be less than obvious, depending on the design. If new batteries don't solve it, then it probably is new thermostat time (had that happen with one of them after 20 odd years of trouble-free service...)

Generally plan on replacing them once a year rather than waiting for them to leave the house cold. As a new homeowner, if you have not had your heating system professionally serviced annually, you might schedule a service call (for the heating system maintenance, not just for getting into the thermostat) and have the service person show you how to get to the batteries, as they will probably have a good idea how to get to them without breaking the thermostat (not uncommon to have to pull it off the wall to get to the batteries, but HOW you pull it off the wall without breaking it is not always evident.)

Otherwise, figure out what brand and model thermostat you have and look up the manual on the internet.