Eq. 2 of Ref [1] (which is by the same author as Ref [2], which the OP mentioned in the comments) defines the Gibbs measure for the canonical ensemble \begin{equation} p(H,\beta) = q(x) \exp\left[-\beta H(x)-W(\beta)\right] \end{equation} where $x$ ranges over configuration space, $H(x)$ is the energy, $\beta=1/kT$ is the inverse temperature, and $W(\beta)$ is a normalization constant.

Let's compare this with the normal way to write Boltzman distribution. The Boltzman distribution gives the probability of finding the system in a given microstate, which is determined by the configuration space variables $x$. \begin{equation} p(x,\beta) =\frac{e^{-\beta H(x)}}{\int dx e^{-\beta H(x)}} = \frac{e^{-\beta H(x)}}{Z} \end{equation} where the partition function is $Z\equiv \int dx e^{-\beta H(x)}$.

It should be understood that $x$ here is not necessarily a single variable, but in general could be a vector of parameters. For instance, $x$ could represent the positions and momenta of $10^{23}$ particles, and in this case $\int dx$ really should be understood as a huge integral over each of these $10^{23}$ configuration space variables.

Now this isn't quite the same thing as Eq 2, which gives the probability of the system having a certain energy $H$, rather than being in a given microstate labelled by $x$.

To convert, we need to include a factor of the density of states $\rho(H)$. This function tells us how many states $\rho(H)d H$ contain an energy from $H$ to $H+dH$.

\begin{equation} p(H,\beta) = \frac{\rho(H) e^{-\beta H}}{Z} = \rho(H) \exp\left[-\beta H - \log Z\right] \end{equation}

Comparing this expression with the first one, we can identify the normalization $W(\beta)$ with the log of the partition function $Z$

\begin{equation} W(\beta) = \log Z \end{equation}

Now it appears that the function $q(x)$ in the high temperature $\beta\rightarrow 0$ limit is related to the density of states \begin{equation} q(x) \equiv \rho(H(x)) \end{equation} To be honest I don't understand the notation $q(x)$, because each microstate should be equally likely in the high temperature limit. So I am not sure if the intention is for the author is intending to allow for a distribution of microstates that violates the assumption that each accessible microstate is equally likely, or if $q(x)$ is meant to include some coarse graining over some subset of microstates, or something else.

References:

[1] https://cds.cern.ch/record/332060/files/9708032.pdf

[2] "Geometrical aspects of statistical mechanics" Physical Review E, 1995, volume 51, number 2.

Eq. 2 of Ref [1] (which is by the same first author as Ref [2], which the OP mentioned in the comments) defines the Gibbs measure for the canonical ensemble \begin{equation} p(H,\beta) = q(x) \exp\left[-\beta H(x)-W(\beta)\right] \end{equation} where $x$ ranges over configuration space, $H(x)$ is the energy, $\beta=1/kT$ is the inverse temperature, and $W(\beta)$ is a normalization constant.

Let's compare this with the normal way to write Boltzman distribution. The Boltzman distribution gives the probability of finding the system in a given microstate, which is determined by the configuration space variables $x$. \begin{equation} p(x,\beta) =\frac{e^{-\beta H(x)}}{\int dx e^{-\beta H(x)}} = \frac{e^{-\beta H(x)}}{Z} \end{equation} where the partition function is $Z\equiv \int dx e^{-\beta H(x)}$.

It should be understood that $x$ here is not necessarily a single variable, but in general could be a vector of parameters. For instance, $x$ could represent the positions and momenta of $10^{23}$ particles, and in this case $\int dx$ really should be understood as a huge integral over each of these $10^{23}$ configuration space variables.

Now this isn't quite the same thing as Eq 2, which gives the probability of the system having a certain energy $H$, rather than being in a given microstate labelled by $x$.

To convert, we need to include a factor of the density of states $\rho(H)$. This function tells us how many states $\rho(H)d H$ have an energy in the interval from $H$ to $H+dH$.

\begin{equation} p(H,\beta) = \frac{\rho(H) e^{-\beta H}}{Z} = \rho(H) \exp\left[-\beta H - \log Z\right] \end{equation}

Comparing this expression with the first one, we can identify the normalization $W(\beta)$ with the log of the partition function $Z$

\begin{equation} W(\beta) = \log Z \end{equation}

Now it appears that the function $q(x)$ in the high temperature $\beta\rightarrow 0$ limit is related to the density of states \begin{equation} q(x) \equiv \rho(H(x)) \end{equation} To be honest I don't understand the notation $q(x)$, because each microstate should be equally likely in the high temperature limit. So I am not sure if the author is intending a distribution of microstates that violates the assumption that each accessible microstate is equally likely, or if $q(x)$ is meant to include some coarse graining over some subset of microstates, or something else.

References:

[1] https://cds.cern.ch/record/332060/files/9708032.pdf

[2] "Geometrical aspects of statistical mechanics" Physical Review E, 1995, volume 51, number 2.