The 2008 world champion has obtained a special VIP pass for the animal, complete with a photograph of the dog wearing trendy headphones, but abandoned plans to bring the pet to last month's Bahrain Grand Prix because Roscoe was reported to be feeling under the weather.
The dog has acquired cult status for some Hamilton fans, with the Mercedes driver regularly posting photographs of him on Twitter.
Numerous 'fake' Twitter accounts have also sprung up in the animal's name.
"He's here. He went to the States with me and he came back the other day," Hamilton, who has a transatlantic relationship with American pop singer Nicole Scherzinger, told reporters at the Circuit de Catalunya when asked about Roscoe's F1 debut.
The Briton said his pet was limping, and had been to the vet, but was recovering after doing the 'splits' while left in the care of his father and stepmother when Hamilton was at the Mercedes factory recently.
"For some reason he's just always in the wars but he's much better today," said the Briton. "He's down in my motorhome, I didn't bring him in."
(Reporting by Alan Baldwin; Editing by John Mehaffey)