There are two different situations to deal with:
A formula or quantity which involves a specific temperature.
- A formula which involves a specific temperature, such as the ideal gas law,$$PV=nRT$$ or the average energy per molecule in a gas,$$< E>=(3/2)kT$$
- A calculation which involves a temperature difference such as heat to cause a temperature change, $$Q=mc\Delta T.$$
In case #1, you must use an absolute scale, either Kelvin or Rankine. Then the $R$ or $k$ must be converted properly, using the size of the temperature unit. In that case, 1 K (unit) = 1.8 R (unit).
In case #2, you can use any scale you want as long as both temperatures ($\Delta T = T_{\mathrm{final}}-T_{\mathrm{initial}}$) are on the same scale, and you use the specific heat, $c$, with the proper temperature unit size, 1 K = 1 C$^o$ = 1.8 F$^o$ = 1.8 R$^o$.
For example, $c$ = 0.7 $\frac{\mathrm{cal}}{\mathrm{g}\cdot\mathrm{C}^o}$ =0.7 $\frac{\mathrm{cal}}{\mathrm{g}\cdot\mathrm{K}}$ = 0.389 $\frac{\mathrm{cal}}{\mathrm{g}\cdot\mathrm{F}^o}$ = 0.389 $\frac{\mathrm{cal}}{\mathrm{g}\cdot\mathrm{R}^o}$