In water, networks of semiflexible fibrils of the protein α-synuclein stiffen significantly with increasing temperature. We make plausible that this reversible stiffening is a result of hydrophobic contacts between the fibrils that become more prominent with increasing temperature. The good agreement of our experimentally observed temperature dependence of the storage modulus of the network with a scaling theory linking network elasticity with reversible cross-linking enables us to quantify the endothermic binding enthalpy and estimate the effective size of hydrophobic patches on the fibril surface. Our findings may not only shed light on the role of amyloid deposits in disease conditions, but can also inspire new approaches for the design of thermoresponsive materials.