  For atoms or molecules in optical lattices, conventional thermometry methods
are often unsuitable due to low particle numbers or a lack of cycling
transitions. However, a differential spectroscopic light shift can map
temperature onto the line shape with a low sensitivity to trap anharmonicity.
We study narrow molecular transitions to demonstrate precise frequency-based
lattice thermometry, as well as carrier cooling. This approach should be
applicable down to nanokelvin temperatures. We also discuss how the thermal
light shift can affect the accuracy of optical lattice clocks.
