As a fan of Pekka Hämäläinen's work I definitely recommend his book "The Comanche Empire", it's an excellent book and it is to my knowledge very well received with some light criticism. One of the more critical reviews that I've read of the book come from The Hispanic American Historical Review where Charles Cutter wrote in his review

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"There is much to commend in this formidable study, and those interested in eighteenth- and nineteenth-century borderlands, Mexican, and U.S. history will want to

read and consider the work. Specialists may quibble with Hämäläinen’s characterization

of Comanche political organization as an “empire”; they may doubt that Spain’s longstanding imperial aim was one of continual expansion, as the author claims, or that Indian

policy in New Spain’s far north was so neatly “top-down” and imperial in scope; and they

may feel that the struggles of the young Mexican nation are given short shrift. Even

nonspecialists may find some of the author’s assertions to be a bit overblown—a product,

perhaps, of his Comanche-centeredness. The Comanche arrival on the southern Plains,

for example, is deemed “one of the key turning points in early American history” (p. 18).

Comanche expansion past the Llano Estacado to the Balcones Escarpment in Texas is

touted as “one of the most explosive territorial conquests in North American history”

(p. 55); and the racial complexity and ambiguity resulting from Comanche slavery is

regarded as the “crucible which forged Anglo-American understandings of Mexicans as

a mixed, stigmatized, and subordinate class” (p. 359). In a similar vein, Hämäläinen tends

to portray Comanchería as the primary focal point of imperial and national administra tors, largely ignoring other, greater concerns that may have occupied their attention.

The Spanish monarchy faced uprisings on the peninsula and in the colonies (including

a full-blown Andean rebellion), Napoleon’s occupation of the Iberian Peninsula, and

colonial wars of emancipation. The fledgling Mexican nation had to deal with the delicate matters of state building and economic resuscitation after independence. And in the

United States, slavery, immigration, and the looming sectional crisis were serious issues.

Efforts to control Comanchería from the outside may have failed, but if looked at from a

variety of other “centers,” Indian policy in what ultimately became the American Southwest, while important, was not the only thing to worry about.

These points aside, Pekka Hämäläinen succeeds admirably in explaining the internal logic and coherence of Comanche power during the eighteenth and nineteenth

centuries and, in doing so, provides a new way to understand the historical development of the greater Southwest. Rubbing against the grain of traditional scholarship, this

Comanche-centered interpretation of events is bound to shake things up. The greater

Southwest will never quite look the same."

another review from American Nineteenth Century History. Sep2011, Vol. 12 Issue 3 written by Paul Mapp reads

"Comanche Empire does not claim to offer new material from some substantial but hitherto undiscovered trove of documents, to push the linguistic boundaries of Anglo-American historians beyond a fairly small number of Spanish-language sources and secondary works, or to introduce a novel historical method. Instead, it re-examines, re-interprets, and re-designates. In the central parts of each chapter, Hämäläinen combines insights from the many strong works of Comanche specialists with conclusions drawn from his own analysis of Comanche actions, southern plains environmental imperatives, and the testimony of authors observing the Comanches. Throughout, he tries not simply to present a chronological account of the rise and demise of one Indian nation, but also to see the southwest and Comanche actions as the Comanches themselves might have, difficult though this may be in light of the inevitable dependence on the writings of Euro-American outsiders. These cogent, learned, and often inspired chapter cores are the backbone of the book. Around them, in the introductions and conclusions of each chapter and of the book as a whole, Hämäläinen pushes the interpretations of his primary and secondary material as far as – in my view, often farther than – evidence and reason will allow."

Both of these reviews, while critical are overall positive of Hämäläinen's work and showcase that while not a flawless book it is definitely worth the time of anyone wanting to learn about Comanche history.

The issue with comparing a work like "The Comanche Empire" to a book like "Empire of the Summer Moon" is that while "The Comanche Empire has been extensively reviewed by academic historians in related fields, "Empire of the Summer Moon" has not been subject to the same level of academic scrutiny and actual academic book reviews of it are basically non-existent.

In terms of reviews of the book this article from the Huffington Post written by the co-founder of the Lakota Times is extremely critical of the book https://www.huffpost.com/entry/book-review-by-tim-giago_b_4628242 . And while the article is primarily focused on "The Heart of Everything That Is" it also levies criticism on "Empire of the Summer Moon". Much criticism of the book largely centers on Gwynne's overemphasis on violence and failure to contextualize the actions of the Comanche in a way that some American Indians would describe as racist.

So in short I recommend reading "The Comanche Empire" it's a great book, and honestly as far as academic history books go it is very reasonable. If you'd like to know a bit more about the book I recommend listening to this interview that Pekka did for the New Books Network https://newbooksnetwork.com/pekka-hamalainen-the-comanche-empire-yale-up-2008 . However, I'd be a bit skeptical of "Empire of the Summer Moon" and be well of aware of the criticism of the book if you do plan on reading it.

Sources:

Mapp, PaulW. “The Comanche Empire.” American Nineteenth Century History, vol. 12, no. 3, Sept. 2011, pp. 351–355.

CUTTER, CHARLES R. “The Comanche Empire.” Hispanic American Historical Review, vol. 89, no. 4, Nov. 2009, pp. 726–728