While “rest” furthers the metaphor of sleep, it necessarily describes the oblivion of death as well. The notion that death might be “rest” offers a positive perspective on the speaker’s eventual fate. Indeed, in Sonnet 73 the speaker takes a resigned, rather than combative, stance against his primary foes—time and death.
“Boughs” here means the limbs of a tree. In Early Modern England it would have evoked connotations of the gallows or the branches of a family tree. “Cold” is a reference to winter, the season that the speaker has used throughout the sequence as a metaphor for the final years of one’s life and ultimate death. In this metaphor, the speaker claims that one’s lineage, or legacy, is threatened by death. He creates a somber tone that will dominate the rest of the poem.