It's difficult to say much about this particular book without giving away the overall plot and outcome, so I'm putting spoiler warnings up in lights here.
... if that is not a concern.
Double Star is one of four novels to have won Hugo Awards for Heinlein. The others were Starship Troopers, Stranger in a Strange Land, and The Moon is a Harsh Mistress. Whatever their particular faults, all four are among the classics of the science fiction genre. They were published over a period of about a decade (1956 for Double Star to 1966 for The Moon is a Harsh Mistress), and this was the period when Heinlein was a truly dominant force in SF. For those familiar with his work, think how diverse these four books (and others published during the same period) are.
While the title might suggest engagement with vast astronomical objects, that is misleading - as are some of Heinlein's other book titles, such as The Star Beast. Double Star is about an actor who is called on to impersonate a politician who has been kidnapped, and who, after being released, suffers serious illness. The words "double star" can be read in various ways, but of course the actor thinks of himself as something of a "star", and he is a double for the politician. There is plenty of room to take this idea further, though I don't know exactly what Heinlein and his publishers had in mind. At any rate, even when we're in the know, there's a small frisson of pleasure in the contrast between what might be immediately connoted by such a title and what the book actually gives us.
The story is narrated in the first person by the actor - Lawrence Smith, also known as the Great Lorenzo - who is called on to play the role of political leader John Joseph Bonforte. At least on its face, the book uses first-person narration in the classic way: to present the story of someone who has survived life-changing experiences and is now reflecting on them, as well as merely recounting them, from a position of greater knowledge and wisdom than when he began.
Such narratives can also, notoriously, raise questions about the reliability of narrators, how much they have really learned, and so on. Often, an author can give us subtle, or not so subtle, clues about a narrator's unreliability. First-person narratives can also be ambiguous or unstable, insofar as the narrator may actually have learned, and yet we're given intimations of the limits to what has been learned, and that he or she has a (possibly long) way to go.
There are interpretations of Double Star along these more complex lines, but I'm somewhat resistant to them. I just don't think the book is all that complicated, much as it is intelligent and well executed. The narrator starts out as a clever, but also vain and shallow, man who eventually matures - becoming trapped in his role, but also obtaining a degree of wisdom and depth, and certain satisfactions (not least, ultimately marrying the woman who was in love with Bonforte when the story began). Most of the events portrayed take place during a relatively short period in Lawrence's life when he made the switch to (more or less) becoming Bonforte. However, Bonforte dies, the switch becomes permanent, and it becomes clear at the end of the novel that Lawrence is telling us the story from a perspective twenty-five years later. That's probably complicated enough.
Interestingly, there is little in the action that requires a futuristic scenario. A very similar story could have been told involving only events and political machinations on Earth and in the present. I.e., the story could have been a mainstream political and psychological thriller with little adjustment.
True, the political intrigue involves relations between humans and extraterrestrial aliens, but this is quite explicitly analogous to the relations of imperialist Europeans and colonised peoples. Part of Bonforte's political platform is anti-imperialist, in that he favours expansion into space in a way that involves some kind of inclusivity among all intelligent life forms, rather than a kind of speciesist human imperialism.
However, this is secondary to the story's focus on Lawrence's predicament and how he responds to it. Really, almost any vaguely idealistic political viewpoint would have been just as effective for the purpose: all that's needed is that Bonforte's views and (vaguely evoked) policies not be ones that readers would find repellent. If they had been, that would have created a very different and more sombre book (think, perhaps, of Kurt Vonnegut's Mother Night). As a result, Double Star does, indeed, seem to endorse Bonforte's ideals, and in doing so it contains something of an anti-imperialist message, but at the same time there is a sense that this is not what is important; it is not what is really going on, thematically. We have more a story about individual identity and how it can change, even radically. Again though, it seems to change for the better in Lorenzo's case, making this seem an optimistic story even if its apparent that the narrator has been guided and shaped largely by forces beyond his control.
Heinlein writes with great efficiency, as usual. By this point of his career, in the 1950s, he had mastered many techniques for moving a story along and dealing with challenges. One is his ability to create brief set-piece debates where one character is always a step ahead of the other - and manages to push the other to accept some viewpoint that initially appears odd or unreasonable. Heinlein is skilled at dealing with artistic problems, or potential objections, by using this formula; thus, he resolves difficulties on stage (the theatrical metaphor is especially appropriate for this book) while also getting them out of the way swiftly.
The technique also enables him to show Lorenzo's growth: in the beginning, he is the one who gets bamboozled and browbeaten by characters who are a step ahead of him, but as the action settles down he is increasingly the one who gains a deeper understanding of situations, and soon he is bamboozling and browbeating others.
Double Star is highly regarded by Heinlein fans: in my experience, some prefer it to more famous books by Heinlein, such as the other three novels that won Hugos. Despite its thematic complications, it is, as I've suggested, not all that complicated, and it ends up being a good, rollicking, amusing yarn. There is a simple movement of events: from the beginning when the narrator is asked to impersonate Bonforte to the implied vantage point from which the story is being narrated (a vantage point that is made explicit only in the final few pages).
All in all, this is a lovely little (an ordinary fast reader could whip through it in a couple of hours; I am more distractable, but it didn't take me long) book that deserves its continued classic status. Its main interest, however, does not especially depend on the science fiction elements. It is interesting to see how they do and don't play a role (as it were), and, conversely, to observe how Heinlein dealt with this kind of thematic and psychological material in an SF setting.