His baptism transcibed by me from an image of the parish register:

[  Camden  >  St George, Bloomsbury  >  1775-1812

“Born                                   1811                     Baptized
“1811                                                              1812
“. . . .
“Augt. 16  Robert James son of George and Margaret Brown    Jany. 24”
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There is no Robert James Brown who is in FreeBMD for dying in 1839, though there are loads of plain Roberts.

Nor can any will be found by him - on TNA.

Nor has any burial been found.

So, currently, no public record for his death.
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Mar, 2021, TFPL: However there is this burial record which could just be his, though it is before the Brown record of his death date, 23 Sep 1839, which was:

“ BURIALS in the parish of Ockley in the Countey of Surrey in the Year 1839
“. . . .
“ Name           Abode       When buried     Age        Celebrant
“. . . .  
“Robert Brown   Abinger       Mar 3rd         28        J Cook
“321
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And this could be the registration for the above burial:

Name:              Age at Death (in years):  
BROWN, ROBERT      28  
GRO Reference: 1839  M Quarter in DORKING  Volume 04  Page 53
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I wonder, did he die outside England and Wales?

FindMyPast, Ancestry, FamilySearch, Burials have all been searched and no “Robert James Brown” and jumped up to say he was there!  There are far too many Robert Browns to begin to guess.
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March 2021, TFPL: Finally, found his burial, confirming that the list had been guessed wrongly!

He did not die until the 1850s, in fact in Sept 1856.

He is buried in Old Brompton Cemetery, transcribed by me from an image of the entry in the register:

“REGISTER OF BURIALS IN THE WEST OF LONDON AND WESTMINSTER CEMETERY, EARL’S COURT, OLD BROMPTON
ESTABLISHED BY ACT OF PARLIAMENT, 1ST VICTORIA, CAP. 130.

Name    Death Place  Burial Date  Age  Celebrant  Grave Location  Parish Of Origin  Church  Remarks

Robert  23 Saville   1857 Sep 27  45   Rev J.G.   Private grave   St James          church  No
James   Row, St.                  yrs  Belling-   I115.6X151.0    Westminster               extra
Brown   James                          ham                                                  depth
14165
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And, looking at him again a few weeks later, I found his entry on the new GRO site with ages at death, he being the only 45-year old Robert James Brown who died from 1854 to 1858 inclusive:

Name:                   Age at Death (in years):  
BROWN, ROBERT JAMES     45  
GRO Reference: 1856  S Quarter in SAINT JAMES WESTMINSTER  Volume 01A  Page 248
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First name(s)           Robert James
Last name               Brown
Burial date             27 Sep 1856
Residence               23 Saville Row, St James
Print number            13738
Burial register number  14165
Piece description       Brompton Cemetery burial book no. 51
Records year range      1856-1856
Archive reference       WORK 97/259
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