Life is a non-equilibrium system, which is nowadays maintained by a highly developed energy conversion machinery. Four billion years ago, other non-equilibrium mechanisms were needed to kick-start living processes. We propose ubiquitous heat fluxes as suitable driving force: Thermal gradients across water filled pores lead to a concurrent fluid convection and directed movement of dissolved charged molecules along the temperature difference. Combined, both effects accumulate the dissolved biomolecules in a length dependent manner. Oligonucleotides are pushed into a hydrogel phase, depending on their sequence and chirality: A mixture of strands with different sequence demixes into sequence-pure and homochiral hydrogels upon thermal accumulation, possibly selecting for interacting strands during the origin of life. The thermal non-equilibrium also creates and maintains a pH gradient over two units by the selective accumulation of charged buffer molecules, which shifts the local equilibrium in pH. In this system, early compartments of life may have cycled between different external pH conditions, implementing an important boundary condition for a primordial metabolism.