H2O2 has been recognized as an important signaling molecule in plants. We sought to establish a genetically encoded, fluorescent H2O2 sensor that allows H2O2 monitoring in all major subcompartments of a Chlamydomonas cell. To this end we engineered the hypersensitive H2O2 sensor, roGFP2-Tsa2ΔCR, as a genetic part for the Chlamydomonas Modular Cloning toolbox. Using previously generated parts, together with new ones, we constructed modules and devices that target the sensor to the cytosol, nucleus, mitochondrial matrix, chloroplast stroma, thylakoid lumen, and ER. The sensor was functional in all compartments, except for the ER where it was fully oxidized. Employing our novel sensors, we show that H2O2 produced by photosynthetic linear electron transport (PET) in the stroma leaks into the cytosol but only reaches other subcellular compartments if produced under non-physiological conditions. Our results thus imply the establishment of steep intracellular H2O2 gradients under normal physiological conditions and suggest that the cytosolic complement of H2O2 scavenging enzymes effectively limits H2O2 diffusion. Furthermore, in heat stressed cells, we show that cytosolic H2O2 levels closely mirror temperature up- and downshifts and are independent from PET. We anticipate that these sensors will greatly facilitate future investigations into H2O2 biology in algal and plant cells.