This is only a partial answer to your question; I believe there is more current work, and have forwarded your question to someone working in this area to see if they have more recent results.

In Theorem 5 of

<cite authors="Bourgain, Jean; Dilworth, Stephen; Ford, Kevin; Konyagin, Sergei; Kutzarova, Denka">_Bourgain, Jean; Dilworth, Stephen; Ford, Kevin; Konyagin, Sergei; Kutzarova, Denka_, [**Explicit constructions of RIP matrices and related problems**](https://arxiv.org/abs/1008.4535), Duke Math. J. 159, No. 1, 145-185 (2011). [ZBL1236.94027](https://zbmath.org/?q=an:1236.94027).</cite>

it is shown (in your notation) that $|2A| \geq |A|^{2\tau}$ for $A \subset [0:d]^n$, where $\tau$ solves the equation
$$ (\frac{1}{d+1})^{2\tau} + (\frac{d}{d+1})^\tau = 1.$$
They do not believe this result to be sharp, and (as you do) conjecture that $|2A| \geq |A|^{\log_{d+1}(2d+1)}$ instead.  This is known for $d=1$, see

<cite authors="Woodall, D. R.">_Woodall, D. R._, [**A theorem on cubes**](https://doi.org/10.1112/S0025579300008913), Mathematika, London 24, 60-62 (1977). [ZBL0349.05010](https://zbmath.org/?q=an:0349.05010).</cite>

From the method of compressions one may assume without loss of generality that $A$ is a downset.  If one then restricts $2A$ to the set $[0:d]^n$ then a sharp answer to your question was worked out in

<cite authors="Bollobás, Béla; Leader, Imre">_Bollobás, Béla; Leader, Imre_, [**Sums in the grid**](https://doi.org/10.1016/S0012-365X(96)00303-2), Discrete Math. 162, No. 1-3, 31-48 (1996). [ZBL0872.11007](https://zbmath.org/?q=an:0872.11007).</cite>

however I do not see an easy way to pass from this restricted sumset problem to the full sumset problem.

There are related results in

<cite authors="Matolcsi, Dávid; Ruzsa, Imre Z.; Shakan, George; Zhelezov, Dmitrii">_Matolcsi, Dávid; Ruzsa, Imre Z.; Shakan, George; Zhelezov, Dmitrii_, [**An analytic approach to cardinalities of sumsets**](https://doi.org/10.1007/s00493-021-4547-0), Combinatorica 42, No. 2, 203-236 (2022). [ZBL1513.11022](https://zbmath.org/?q=an:1513.11022).</cite>

and the analogous problem for energies was studied in [this recent paper of de Dios, Greenfeld, Ivanisvili, and Madrid][1], which would give some lower bounds on $|2A|$, but probably not the optimal ones.


  [1]: https://arxiv.org/abs/2112.09352