The earliest reference to some aspect of
this expression goes all the way back to the Ancient Egyptians. They
noted that the heliacal rising of the star Sirius heralded the
hottest part of the summer. The star’s hieroglyph is a dog. Sirius
would appear in Egypt just before the season where the Nile
typically floods. So it is thought the star’s hieroglyphic symbol
being a dog symbolized a “watchdog”.
It is the brightest star in what is now known as the Canis Major
(Latin for Greater Dog) constellation. It’s rising marked the start
of the hottest part of the year, which then became the 'Dog Days'.
The Roman’s and Greeks had expressions for Dog Days. They both
believed that, when Sirius rose around the same time as the Sun,
this contributed to that time of year becoming hotter. As such, they
would often make sacrifices to Sirius, including sacrificing dogs,
to appease Sirius with the hope that this would result in a mild
summer and would protect their crops from scorching. Seems to me
that offering dead dogs to a dog might not please him as much as
they thought.