Temperature

Thermometers wouldn't work if energy didn't spontaneously flow from two objects in contact with each other, and temperature is objectively, just something you measure with a thermometer. (of course, we're talking about traditional mercury thermometers here, and not stuff like infrared thermometers)

One definition for the average kinetic energy of an ideal gas (with three degrees of freedom) is $$\bar K=\frac{3}{2}k_BT$$ and we can get a bad definition of temperature by moving the terms around. Note that we got this equation from the equipartition theorem which states the average energy of a system in which individual elements have $n$ degrees of freedom is $\displaystyle\frac{1}{2}nk_BT$.

Fundamental Assumption of Statistical Mechanics: All accessible microstates are equally likely for a system in thermodynamic equilibrium.

Entropy is a statistical measure of how many possible arrangements exist for a system. Define multiplicity $\Omega$ for an Einstein Solid with $N$ quantum harmonic oscillators and $q$ units of energy as $$\Omega={q+N-1\choose q},$$ and energy units will flow in such a way as to maximize the product of individual multiplicities of the objects in contact with each other. Concretely, there are systems $A,B$ with $N_A,N_B$ oscillators and $q_A,q_B$ units of energy in contact with each other. They have individual multiplicities of $\Omega_A,\Omega_B$ and the units of energy $q_A,q_B$ will flow in a way that maximizes $\Omega=\Omega_A\cdot\Omega_B.$ The constraint obviously being that $q_A+q_B$ is always fixed. The graph of $\Omega$ versus $q_A$ looks like a Dirac Delta function with a very narrow peak near the middle, which is intuitive (if you accept that multiplicities should be maximized, because the $\displaystyle{n\choose k}$ function has it's maximum at $2k=n$)

Now, entropy is just the natural log of this multiplicity and has units [energy]$\times$[temperature]$^{-1}$.$$S=k_B\ln{\Omega}$$ The Second Law of Thermodynamics: (1) The entropy of an isolated system tends to increase. (2) A system in thermodynamic equilibrium is most likely to be found in the macrostate (configuration of $q_A$ and $q_B$ for example above) of highest entropy.

Finally, temperature is defined as $$\frac{1}{T}=\frac{\Delta S}{\Delta E}.$$ Which is why temperature is a measure of the tendency of an object to give up it's energy to another object it comes in contact with. Reference/Inspiration: Temperature as Joules per Bit