There could be some gunk left in the carburetor somewhere. I am having success with adding gumout or something similar to the gas tank on our mower, typically once a year. You have to operate the engine for a half-hour or more for the cleaning solvent to have much effect.

It's the brass bolt on the bottom of the fuel bowl (float bowl). Tip the mower up on it's right-side wheels so the carb is up. Remove the bolt and the bowl should stay in place. Clean the bolt and it's apertures, including one you might normally miss - it's a black speck at the top at the end of the threads. Take a wire tie like from a bread wrapper, strip the paper off and use it to rod out this bleed that goes down at an angle. Blast out with carb cleaner. Replace bolt and snug down, don't over-tighten. Set the mower down and check for a leak at the bolt. If it starts leaking, give it a 1/8 turn with your wrench - repeat if needed.
This should get the mower back to work. You can forestall future problems by using a fuel stabilizer/fuel system cleaner like Sta-Bil or Seafoam all the time. Put it in your gas can when you buy gas. At the end of the season, you don't have to do anything with the fuel- just do the other year-end things for storage.

I would suspect the jet if he hadn't mentioned that it runs fine when it's warmed up. Jets still remain clogged when the motor is hot.

It's a strange problem.

Shown are 3 of the typical jets and the small holes. All 4 are Tecumseh jets. Briggs jets are similar with the center-bottom hole and one on the side. All these holes need to be fully clean and open for the engine to run correctly.
Some jets only have a small hole in the side as the old style Tecumseh jet shown.

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It is indeed kind of weird. These engines have no cold-enrichment other than the primer, so if the bleed were only partially blocked, I can see how it would possibly make for lean-running during cold operation but allow just enough fuel flow for hot operation.

Thanks to all for the assistance. It was the tiny jet just like several of you mentioned. It was so small I didn't even notice it when I cleaned the nut. Anyway, this weekend I pulled the nut, leaving the carburetor bowl in place, and cleaned it with a wire. It took all of ten minutes and now the lawnmower runs fine. I appreciate the assistance.

Ba-da-bing!