Is the Gibbs standard free energy always constant?
In short, no, the standard Gibbs free energy change is not constant; it is a function of temperature. The same is true for practically all other standard-state quantities.

This gets a little confusing because of how standard-state properties are often explained in lower-level (high-school and college-freshman level) textbooks. The standard Gibbs free energy change, $\Delta G^\circ$, is the value of $\Delta G$ at a certain constant pressure (usually 100&nbsp;kPa or 1&nbsp;atm) and constant concentration of solutions (usually 1&nbsp;molal or 1&nbsp;molar) and a specified temperature (usually $0~^\circ \text{C}$ or $25~^\circ \text{C}$). What is intended but not immediately obvious is that the pressure and concentration are constants but temperature is a variable; since the temperature is a variable, $\Delta G^\circ$ is a function of temperature. When the value of $\Delta G^\circ$ is given, the temperature must also be specified.

In your equation, then, $\Delta G^\circ$ is the standard Gibbs free energy change of the reaction at the specified temperature ($T$ in the equation). Change $T$, and the value of $\Delta G^\circ$ also changes; but it is still the standard Gibbs free energy change at the (new) specified temperature. (This equation cannot be used to calculate $\Delta G^\circ$ at a given temperature because  $K_\text{eq}$ is also a function of temperature.)

The IUPAC "Green Book" (ed.&nbsp;3) has a good explanation of what is meant by the term "standard state"; the discussion starts on p.&nbsp;61 (as printed on the page; it's p.&nbsp;76 in the pdf file). It explicitly expresses the standard chemical potential ($\mu^\circ$), which is very closely related to $\Delta G^\circ$, as a function of temperature.