The entropy of a probability distribution $\rho$ is:
$$ S[\mathbb{Q},\rho]=-k_B \sum_{q\in\mathbb{Q}} \rho[\mathbb{Q},q] \ln \rho[\mathbb{Q},q] \tag{1} $$
Let the following be constraints:
$$ \overline{E}_1 = \sum_{q\in\mathbb{Q}} E_1[q] \rho[q]\\ \vdots\\ \overline{E}_n = \sum_{q\in\mathbb{Q}} E_n[q] \rho[q]\\ 1=\sum_{q\in\mathbb{Q}}\rho[q] $$
The probability measure that maximizes the entropy under these constraints is the Gibbs measure:
$$ \rho[\mathbb{Q},q,\beta_1,\dots,\beta_n] = \frac{1}{Z[\mathbb{Q},\beta_1,\dots,\beta_n]}\exp(-\beta_1 E_1[q] - \dots - \beta_n E_n[q]) \tag{2} $$
$$ \text{where }Z[\mathbb{Q},\beta_1,\dots,\beta_n]=\sum_{q\in\mathbb{Q}} \exp (-\beta_1 E_1[q] - \dots - \beta_n E_n[q]) $$
Replacing (2) into (1), we eventually get:
$$ S[\mathbb{Q},\beta_1,\dots,\beta_n]=k_B \ln Z[\mathbb{Q},\beta_1,\dots,\beta_n]+\beta_1\overline{E}_1[\mathbb{Q},\beta_1,\dots,\beta_n]+\dots+\beta_n\overline{E}_n[\mathbb{Q},\beta_1,\dots,\beta_n] $$
and its total derivative is:
$$ dS[\mathbb{Q},\beta_1,\dots,\beta_n]=\beta_1d\overline{E}_1[\mathbb{Q},\beta_1,\dots,\beta_n]+\dots+\beta_nd\overline{E}_n[\mathbb{Q},\beta_1,\dots,\beta_n]\tag{3} $$
I am trying to understand thermodynamics from a strong mathematical footing, but I wasn't able to find any physics textbook that addresses the following issue:
Thermodynamics starts from the fundamental relation (3), appears to ignore the dependence of $\overline{E}_i$ on its variables $[\mathbb{Q}, \beta_1,\dots,\beta_n]$, instead postulates that they are independent variables and then draws conclusions dependant upon this erasure. With the switcharoo, the relation becomes:
$$ dS[\beta_1,\dots,\beta_n,\overline{E}_1,\dots,\overline{E}_n]=\beta_1d\overline{E}_1+\dots+\beta_nd\overline{E}_n $$
As an example, consider the simple case where dS depends only on one thermodynamic pair:
$$ dS[\beta,\overline{E}]=\beta d\overline{E} $$
In thermodynamics, one often claims that the temperature quantities the 'number of times' one must doubling the energy, while maintaining the temperature constant, to doubles the entropy. Other claims, also dependant upon the erasure, are made regarding the ability to draw cycles over the independent variables, etc.
However, inspecting the true dependance of $S$ and $\overline{E}$ one finds that the energy cannot be doubled without changing $\beta$. Explicitly, one must change $\beta$ to change anything at all:
$$ dS[\beta]=\beta d\overline{E}[\beta] $$
Since thermodynamics depends upon these claims, my question is then how do we save thermodynamics using the formalism above as the starting point?
I have explicitly included the dependance on $\mathbb{Q}$ (which is not normally done) as a last-ditch effort to save thermodynamics. The full dependence is actually:
$$ dS[\mathbb{Q},\beta]=\beta d\overline{E}[\mathbb{Q},\beta] $$
Is thermodynamics then saved by 'changing $\mathbb{Q}$ while maintaining $\beta$ constant (so as to double the energy while keeping $\beta$ constant)'? Its the only thing left to work.