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We consider the problem of sequential multiple hypothesis testing with nontrivial data collection costs. This problem appears, for example, when conducting biological experiments to identify differentially expressed genes of a disease process. This work builds on the generalized $α$-investing framework which enables control of the false discovery rate in a sequential testing setting. We make a theoretical analysis of the long term asymptotic behavior of $α$-wealth which motivates a consideration of sample size in the $α$-investing decision rule. Posing the testing process as a game with nature, we construct a decision rule that optimizes the expected $α$-wealth reward (ERO) and provides an optimal sample size for each test. Empirical results show that a cost-aware ERO decision rule correctly rejects more false null hypotheses than other methods for $n=1$ where $n$ is the sample size. When the sample size is not fixed cost-aware ERO uses a prior on the null hypothesis to adaptively allocate of the sample budget to each test. We extend cost-aware ERO investing to finite-horizon testing which enables the decision rule to allocate samples in a non-myopic manner. Finally, empirical tests on real data sets from biological experiments show that cost-aware ERO balances the allocation of samples to an individual test against the allocation of samples across multiple tests.