The expansion is not caused by any heating but simply the momentum caused by the big bang. Note also the expansion is happening to space itself, not just the stars, etc. The example often given is an expanding balloon with dots drawn on the surface. Not only are the dots moving away from each other but also the "space" between the balloon, namely the balloon itself is expanding (obviously). Likewise the space between the galaxies is expanding.
The expansion leads to cooling just like when a gas expands it cools, or conversely when a gas is compressed it heats up.
