Unless you can change this program yourself, the only way is to script. Luckily that is straightforward (on Windows, I assume you can do the same on Linux in a similar way):
:: Get date and remove / character set d=%date:/=% if exist "%TEMP%\%d%" goto :EOF echo _ > "%TEMP%\%d%" start program
EDIT
Right now I have ~30,000 file in my TEMP folder, on a system I don't use too much, so I wouldn't worry too much about 365 1-byte files over a year. Admittedly, these files still take a full block (4kB?), and maybe you're short of diskspace, so here is a second script that only uses a single file.
set today=%temp%\today if exist "%today%" ( find "%date%" "%today%" if errorlevel 0 goto :EOF ) echo %date% > "%today%" start program