Abstract. The mechanical first law (MFL) of black hole spacetimes is a geometrical relation which relates variations of mass parameter and horizon area. While it is well known that the MFL of asymptotic flat black hole is equivalent to its thermodynamical first law, however we do not know the detail of MFL of black hole spacetimes with cosmological constant which possess black hole and cosmological event horizons. Then this paper aims to formulate an MFL of the two-horizon spacetimes. For this purpose, we try to include the effects of two horizons in the MFL. To do so, we make use of the Iyer-Wald formalism and extend it to regard the mass parameter and the cosmological constant as two independent variables which make it possible to treat the two horizons on the same footing. Our extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and gives the abstract form of MFL of black hole spacetimes with cosmological constant. Then, as a representative application of that formalism, we derive the MFL of Schwarzschildde Sitter (SdS) spacetime. Our MFL of SdS spacetime relates the variations of three quantities; the mass parameter, the total area of two horizons and the volume enclosed by two horizons. If our MFL is regarded as a thermodynamical first law of SdS spacetime, it offers a thermodynamically consistent description of SdS black hole evaporation process: The mass decreases while the volume and the entropy increase. In our suggestion, the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process.

The ultimate goal of this research is to construct a new direct CO2 fixation system using photosystems in living algae. Here, we report light-driven formate production from CO2 by using cyanobacterial photosystem I (PS I). Formate, a chemical hydrogen carrier and important industrial material, can be produced from CO2 by using the reducing power and the catalytic function of formate dehydrogenase (FDH). We created a bacterial FDH mutant that experimentally switched the cofactor specificity from NADH to NADPH, and combined it with an in vitro-reconstituted cyanobacterial light-driven NADPH production system consisting of PS I, ferredoxin (Fd), and ferredoxin-NADP+-reductase (FNR). Consequently, light-dependent formate production under a CO2 atmosphere was successfully achieved. In addition, we introduced the NADPH-dependent FDH mutant into heterocysts of the cyanobacterium Anabaena sp. PCC 7120 and demonstrated an increased formate concentration in the cells. These results provide a new possibility for photo-biological CO2 fixation.