Consider an isolated system of $N$ particles in a box of volume $V$ with a total internal energy, $U$. This box has a partition through it which divides the system into two parts — one with $N_1, V_1, U_1$ and the second with $N_2, V_2, U_2$, such that $$N_1 + N_2 = N$$ $$V_1 + V_2 = V$$ $$U_1 + U_2 = U$$
This common setup will allow us to define a variety of ``experiments’’ and consequently understand the various intensive thermodynamic quantities.
Temperature
First consider an experiment where the partition allows for transfer of energy but not of particle or changing of volume. How do we define the new equilibrium on allowing exchange of energy? Now, though $N_1, V_1$ and $N_2, V_2$ are constants, $U_1$ and $U_2$ are not. They are, however, still constrained by $U_1 + U_2 = U$. We have to change the energy of the two compartments until entropy is maximized. Notice that $dU_1 = -dU_2$. So there is only a single independent variable.
$$S = S_1(U_1, V_1, N_1) + S_2(U_2, V_2, N_2)$$ $$\frac{\partial S}{\partial U_1} = \frac{\partial S_1}{\partial U_1} - \frac{\partial S_2}{\partial U_2} = 0$$ $$\frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2}$$So, energy flows from one compartment to the other till these two partial derivatives become equal. Physical intuition tells us that this partial derivative, therefore, must have something to do with the temperature. Let us assume that $\frac{\partial S}{\partial U} = f(T)$ for some function $f$.
Now, let us think about the route to establishment of equilibrium. Over time, the total entropy $S = S_1 + S_2$ must increase. $$\frac{dS}{dt} = \frac{dS_1}{dt} + \frac{dS_2}{dt} > 0$$ $$\left(\frac{\partial S_1}{\partial U_1} - \frac{\partial S_2}{\partial U_2}\right)\frac{dU_1}{dt} > 0$$
This shows us that if $\frac{\partial S_1}{\partial U_1}=f(T_1)>\frac{\partial S_2}{\partial U_2} = f(T_2)$ at the initial time, then $\frac{dU_1}{dt}>0$. This means that the energy is flowing from the second compartment to the first compartment. So, we define $f(T) = \frac{1}{T}$, which satisfies this direction of energy flow.