The recently established resource theory of quantum thermodynamics offers a framework to explore the ultimate possibilities and limitations in the manipulation of quantum states and in the implementation of nanoscale thermal machines. A core endeavour of this programme is to determine under which conditions can a nonequilibrium quantum state be converted into another using thermal operations. Here we settle this question in the important case of Gaussian quantum states and operations. We provide a complete characterisation of Gaussian thermal operations acting on an arbitrary number of bosonic modes, and derive a simple geometric criterion establishing necessary and sufficient conditions for state transformations under such operations in the general single-mode case, encompassing states with nonzero coherence (squeezing) in the energy eigenbasis. Our analysis leads to a no-go result for the technologically relevant task of algorithmic cooling: We show that it is impossible to reduce the entropy of a system coupled to a Gaussian environment below its own or the environmental temperature, by means of a sequence of Gaussian thermal operations interspersed by arbitrary (even non-Gaussian) unitaries. These findings establish fundamental constraints on the usefulness of Gaussian resources for quantum thermodynamic processes.