Horizon Problem: How did different regions of space acquire different temperatures? I'm trying to understand the Horizon Problem.  Inflation describes how non-causally connected parts of the universe (that we see today) could once have been in causal contact.  The reasoning goes that those regions must have been in causal contact to achieve the uniform temperature that we observe and, thus, we need Inflation to explain the uniform temperature of the sky.
If the universe started out with the same energy and volume everywhere, and the volume at every point in space grew at exactly the same rate, then how could one part of space be a different temperature than another?
 A: Thermodynamics is a classical theory that axiomatically should hold wherever it can be applied. As it is also emergent from statistical mechanics the frameworks where its laws apply have been extended to the universe as observed and modeled with the original Big Bang theory, which fitted a lot of the observations. Because General Relativity is involved, mathematically there are regions in space that thermodynamic equilibrium, which is an axiomatic concept, cannot be reached at the early ages of the universe.
The original  Big Bang model predicts that  there should be random discrepancies in the cosmic microwave background, regions that could not interact with radiations and "scatterings" should not display the uniformity observed,  of order of $10^{-5}$ in the data.  This forced a modification to the original Big Bang model

The high degree of uniformity throughout the observable universe and its faint but measured anisotropy lend strong support for the Big Bang model in general and the ΛCDM ("Lambda Cold Dark Matter") model in particular. Moreover, the fluctuations are coherent on angular scales that are larger than the apparent cosmological horizon at recombination. Either such coherence is acausally fine-tuned, or cosmic inflation occurred.

You ask:

If the universe started out with the same energy and volume everywhere, and the volume at every point in space grew at exactly the same rate, then how could one part of space be a different temperature than another?

Because thermodynamic equilibrium cannot be reached when there are no interaction paths. Even if in the original Big Bang there was uniformity at $(0,0,0,0)$ plus an infinitessimal dt, the fact that there are regions with no causal connections and the fact that thermodynamics emerges from statistical mechanics in a probabilistic formulation makes the uniformity highly improbable, and thus the conclusion above:

Either such coherence is acausally fine-tuned, or cosmic inflation occurred. 

As fine tuning is not something that is popular for  physics models , cosmic inflation was invoked to solve the discrepancy.
edit after comments:
Here it is stated that 

Before a time classified as a Planck time, $10^{-43}$ seconds, all of the four fundamental forces are presumed to have been unified into one force. All matter, energy, space and time are presumed to have exploded outward from the original singularity. Nothing is known of this period. 

italics mine
In this link it is stated that there are about $10^4$  acausal regions in the CMB map we have of the observable universe. Nevertheless the black body curve from the CMB is one that fits very well the theoretical expression, at a level of $10-5}$ anisotropy.
Thermodynamics is based on statistical mechanics so there is a  probability for the initial energy even though the same in each present acausal region to develop more or less particles if considering only a GUTS type model with gravitons included.
In addition there is no guarantee that the same number of zero mass particles ( before symmetry breakings)  will be generated from the primordial energy in each acausal region of the $10^4$ . May be more protons are produced in one slice and only neutrinos in another by statistical fluctuation: at symmetry breaking this could make a large difference between energy bound in masses  and kinetic energy. It is kinetic energy that defines temperature. 
Thus either one has to assume acausal fine tuning in the theory, or develop a quantum mechanical model that could homogenize the universe, before the symmetry breaking time  of the GUTS particles. The inflation model seems to be successful in this.
