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In the modified measure formulation the string tension appear as anadditional dynamical degree of freedom and these tensions are not universal,but rather each string generates its own tension, which can have a differentvalue for each string. We consider new background field that can couple tothese strings, the tension scalar is capable of changing locally along theworld sheet the value of the tension of the extended object. When many types ofstrings probing the same region of space are considered this tension scalar isconstrained by the requirement of quantum conformal invariance. For the case oftwo types of strings probing the same region of space with differentdynamically generated tensions, there are two different metrics, associated tothe different strings, that have to satisfy vacuum Einsteins equations and theconsistency of these two equation determines the tension scalar. In a previouspaper we studied solutions that completely avoid singularities, but withpositive and negative tensions strings. In this paper we will consider onlypositive tension strings for the cosmological case and for warped space time .In both of these cases there is a region where the string tensions approachinfinity, in the cosmological case this takes place in the early universe whilein the warped case, for some value of the warping coordinate and since theHagedorn temperature is proportional the the string tension, we get this waystring scenarios with no limiting Hagedorn temperature in the early universeand this opens the possibility of a string cosmology without a Hgedorn phasetransition through all of its history . Similar situation can take place in awarped space time, for this case, assuming the string have a tendency to avoidregions with lower Hagedorn temperature, we obtain a mechanism for condensationof strings into a surface at high temperatures.

Escaping the Hagedorn Temperature in Cosmology and Warped Spaces with Dynamical Tension Strings