For multihost pathogens, adaptation to multiple hosts has important implications for both applied and basic research. At the applied level, it is one of the main factors determining the probability and the severity of emerging disease outbreaks. At the basic level, it is thought to be a key mechanism for the maintenance of genetic diversity both in host and pathogen species. Using Tobacco etch potyvirus (TEV) and four natural hosts, we have designed an evolution experiment whose strength and novelty are the use of complex multicellular host organism as hosts and a high level of replication of different evolutionary histories and lineages. A pattern of local adaptation, characterized by a higher infectivity and virulence on host(s) encountered during the experimental evolution was found. Local adaptation only had a cost in terms of performance on other hosts in some cases. We could not verify the existence of a cost for generalists, as expected to arise from antagonistic pleiotropy and other genetic mechanisms generating a fitness trade-off between hosts. This observation confirms that this classical theoretical prediction lacks empirical support. We discuss the reasons for this discrepancy between theory and experiment in the light of our results. The analysis of full genome consensus sequences of the evolved lineages established that all mutations shared between lineages were host specific. A low degree of parallel evolution was observed, possibly reflecting the various adaptive pathways available for TEV in each host. Altogether, these results reveal a strong adaptive potential of TEV to new hosts without severe evolutionary constraints.