Which of the following relations about $\mathrm{E}^{\circ}$ cell is false?

Determining the false electrochemical relation:

The standard formula for cell potential is $E_\text{cell}^\circ = E_\text{cathode}^\circ - E_\text{anode}^\circ$ where $E_\text{cathode}^\circ$ and $E_\text{anode}^\circ$ represent standard reduction potentials, so option A is correct.

At 298 K, the relation $E_\text{cell}^\circ = \frac{0.0592}{n} \times \log_{10} K$ derives from the Nernst equation at equilibrium and thermodynamic principles, making option B valid.

The thermodynamic relation between Gibbs free energy and cell potential is $\Delta G^\circ = -nFE_\text{cell}^\circ$. Rearranged, $E_\text{cell}^\circ = -\frac{\Delta G^\circ}{nF}$. Option C omits the negative sign and is therefore incorrect.

Option D represents $E_\text{cell}^\circ = E_\text{anode}^\circ + E_\text{cathode}^\circ$ where $E_\text{anode}^\circ$ is interpreted as oxidation potential. Since oxidation potential equals the negative of reduction potential, this simplifies to $E_\text{cathode}^\circ - E_\text{anode}^\circ$, matching option A and making D valid.

Only option C contains a fundamental error in sign convention.

The false relation is $\boxed{C}$.