Wikipedia gives us the basic facts: the ancient Romans called the hottest part of July and August (approximately from July 23 to August 24) the "dies caniculares," or dog days, and associated them with the time the star Sirius rose at approximately the same time as the sun (its heliacal rising). Sirius was called the "dog star" because it appeared in the constellation Canus Major, or the Large Dog. Because of the shift in the earth's rotation on its axis (precession of the equinoxes) over the centuries, this is no longer an exact correspondence. These days have long had a folkloric association with misfortune.
The term "dog days" was also used in ancient Greece. The name Sirius comes from the Greek word seirios, or "burning." In ancient Egypt, the summer solstice and the heliacal rising of Sirius were marked as the New Year, and came shortly before the annual flooding of the Nile. This was so important to the Egyptians, they aligned their temples with the path of Sirius across the sky, and the air shafts within the Great Pyramid opens to the part of the sky in which Sirius can be viewed. Understandable, since the floods deposited the arable silt the Egyptians needed to grow their food crops.
Of course, we all knew that Sirius was called The Dog Star because a) Sirius Satellite Radio once had a dog as part of its logo (it seems to have stopped using the dog image), and b) J.K. Rowling (the artist also known as Robert Galbraith) gave the name Sirius Black to her dog-shapeshifter. "Why Would Sirius Black Become a Black Dog?" is a chapter in David Colbert's The Magical Worlds of Harry Potter.
Blogger VISUP, whom you may remember from "Imbolc, Buddy Holly and Human Sacrifice," wrote a post called "Sirius Rises" that deals with the mythology of the dog days. The post refers to an ancient belief that Sirius and its hot, drying effect on the earth will influence women's sexuality, making women's desire increase and, perhaps, leading women to initiate certain sexual rituals. In addition to sexual rites, the ancient Romans considered the rising of Sirius the appropriate time to sacrifice a brown dog to Demeter. VISUP suggests this custom may have replaced an even more ancient custom of human sacrifice.
VISUP may be suggesting a link - not necessarily cause and effect - between the ancient sacrifices and a modern-day higher rate of violence in the summer months. VISUP only implies this, so I don't know from statistical evidence whether there is more violence in the hot months or in the cold months.
The link between a goddess associated with the Underworld (if we consider Persephone, Demeter and Hecate as three forms of the same Underworld goddess) and dogs is that dogs were considered guardians of the underworld. In Egypt, VISUP tells us, the goddess and god associated with the dog days are Isis and the jackal-headed Anubis, "her guardian dog."
If we return to Colbert, we'll see that the most common places in English folklore associated with the "grim," the spectral black dog central to Harry Potter and the Prisoner of Azkaban, are churchyards and "certain roads" (probably corpse roads). This may be a folkloric echo of the idea of supernatural dogs guarding the Underworld.
The main idea of VISUP's post is that the dog days are an unusually dangerous time of year to this day, essentially that the ancient Greeks and Romans were right to be wary of them, even if the cause of this strangeness (if it, in fact, exists outside of people's subjective perception) wasn't known then and isn't known now.
Another blogger of unusual phenomena, Christopher Loring Knowles of the blog Secret Sun, uses the dog days as a recurring theme. Loring Knowles suggests another link between the goddess and canines: the Egyptians associated the heliacal rising of Sirius with the goddess Isis, and the destructive power of goddesses was often compared in ancient times to predatory animals such as lions (as in the sun-goddess Sekhmet), tigers, wolves and hunting dogs (the Greek goddess Artemis/Roman Diana is sometimes depicted with one or more hunting dogs). He also points out that Demeter's Roman name, Ceres, is similar to Sirius, and implies that Isis, Ceres and Sirius may all be names for one being.
Loring Knowles' main thesis seems to be much the same as VISUP's: that for some unknown reason, strange and unlucky things with some association to dogs and/or Sirius seem to happen during the dog days. A subtextual lesson seems to be that the hottest, strangest days of the summer are associated with a great goddess, with great potential to cause harm - or at least mischief (because not all summer weirdness has life-or-death consequences) - for human beings with an overabundance of heat, both literal and sexual.
How do you feel about the dog days of summer?